How much faith do you need to become a buddhist?

The disciples of the Supreme Buddha are divided into two main categories: laity and clergy (Bhikkhus). These disciples only take refuge in the Triple Gem. They do not possess the weakness of taking refuge in various other views. That is called faith or Saddha. This faith has to be genuine and unflinching.

The Buddha’s disciples are genuinely faithful towards the Triple Gem. They are also wise, brave, and dignified. The disciples of the Buddha do not bow in front of other false views. They do not share the false pretense of grasping every belief as true and slip away from the truth.

Sadly in the present day and age there are many so-called-Buddhists that bear misguided and false views harming themselves and many around them. In Sri Lanka specially, the country of my birth, there are many who carry false beliefs while taking refuge in auspicious times according to asterism, auspicious signs, horoscopes, planetary objects, and the twelve houses of astrology. None of these have anything to do with Buddha’s teachings.

A disciple who has developed the powers of a Noble learner such as faith, virtue, learning, liberality and wisdom, has the qualities to receive a rebirth in those Deva Worlds (being born in the heavens). Such a disciple with the powers of a Noble learner could contemplate: "Devas too have noble qualities that I possess. After my death in this world, I can receive rebirth among the Devas". Thus the disciple with the powers of a Noble learner does the meditation called 'Contemplation of the Devas.' . In the Doctrine, there is a meditation called the 'Contemplation of Devas'. In it the Buddha spoke about The World of the Four, The World of the Thirty Three, Yama, Tusita, Nimmanarati, and Paramimmita Vasavarti. These are all Deva Worlds.

If one cannot lead a life of virtue and Dhamma in the worlds of Devas, The Buddha would not have taught His disciples how to meditate on the 'Contemplation of the Devas'. Another thing we know is that when the Buddha preached 'The Motion of the Wheel of Truth' (Dhamma Cakka Pavattana Sutta) only one person in the world of human beings reached fruition of the path, while Devas in twenty two worlds of Devas reached fruition of the path. This is mentioned in this same sutta.

In Samyukta Nikaya there are discourses called Devata Samyukta and Devaduta Samyukta. When these discourses were spoken by the Buddha, there were many beings from the world of Devas who reached the fruition of the Path. Sakkapanha Sutta in Digha Nikaya (collection of long discourses) shows that Sakra, the Lord of the Deva world of the Thirty Three had reached fruition of the Path. So does one need any more evidence than that?

Buddhist philosophy is built on facts and not on faith. Once you studied facts carefully and question your self and practice the teaching you will see it through yourself.

It depends on individual. Some starts with small faith ends with big faith. Some starts with big faith ends with little faith.

An accomplished Buddhist practitioner has unwavering faith no matter what happens.

In hardcore ('supramundane') Buddhism, it is taught the cause for the generation of faith is your personal experience of suffering or dissatisfaction with ordinary worldly life. To quote:

Faith, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for faith? 'Suffering' should be the reply.

Upanisa Sutta

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And what is the noble search? There is the case where a person, himself being subject to birth, seeing the drawbacks of birth, seeks the unborn, unexcelled rest from the yoke: Nibbana. Himself being subject to aging... illness... death... sorrow... defilement, seeing the drawbacks of aging... illness... death... sorrow... defilement, seeks the aging-less, illness-less, deathless, sorrow-less, undefiled, unexcelled rest from the yoke: Nibbana. This is the noble search.

Ariyapariyesana Sutta

In addition, the scriptures explain hearing the True Teaching is a cause of faith.

Faith, too, has its nutriment; it is not without a nutriment. And what is the nutriment of faith? 'Listening to the True Teaching,' should be the answer.

AN 10.61

The hardcore scriptures explain individual people are drawn to eachother due to the individual genetic (dhatu) composition of their bio-psychology (nama-rupa). For certain individuals, hearing about unknowable things in Buddhism will turn them away from Buddhism therefore the purpose of Buddhism will never be achieved by anyone on the planet Earth (since craving for future lives, similar to how many Christians are interested in living forever, is contrary to the path to Nibbana).

Therefore, on the hardcore level, faith in external hells and heavens, karmic-rebirth, other realms etc, is not required, which is why in the 1st three sermons attributed to the Buddha, which gave rise to many enlightened beings, these things were not mentioned at all.

If you say that you can see them by yourself, yes, in hardcore Buddhism, this is considered to be an illusion like when we have a lucid dream. To quote:

Now suppose that a man desiring heartwood, in quest of heartwood, seeking heartwood, were to go into a forest carrying a sharp ax. There he would see a large banana tree: straight, young, of enormous height. He would cut it at the root and, having cut it at the root, would chop off the top. Having chopped off the top, he would peel away the outer skin. Peeling away the outer skin, he wouldn't even find sapwood, to say nothing of heartwood. Then a man with good eyesight would see it, observe it, & appropriately examine it. To him — seeing it, observing it, & appropriately examining it — it would appear empty, void, without substance: for what substance would there be in a banana tree? In the same way, a monk sees, observes, & appropriately examines any mental fabrications that are past, future, or present; internal or external; blatant or subtle; common or sublime; far or near. To him — seeing them, observing them, & appropriately examining them — they would appear empty, void, without substance: for what substance would there be in fabrications?

"Now suppose that a magician or magician's apprentice were to display a magic trick at a major intersection, and a man with good eyesight were to see it, observe it, & appropriately examine it. To him — seeing it, observing it, & appropriately examining it — it would appear empty, void, without substance: for what substance would there be in a magic trick? In the same way, a monk sees, observes, & appropriately examines any consciousness that is past, future, or present; internal or external; blatant or subtle; common or sublime; far or near. To him — seeing it, observing it, & appropriately examining it — it would appear empty, void, without substance: for what substance would there be in consciousness?

Phena Sutta

In conclusion, if the mind can see clearly the illusionary nature of all mental phenomena & sense objects of consciousness, it should be already enlightened (therefore does not require to ask questions about faith in Buddhism).

you don't need faith to become a Buddhist. Buddhism is strictly not a religion in the context of being a faith and worship owing allegiance to a supernatural being.

No saviour concept in Buddhism. A Buddha is not a saviour who saves others by his personal salvation. Although a Buddhist seeks refuge in the Buddha as his incomparable guide who indicates the path of purity, he makes no servile surrender. A Buddhist does not think that he can gain purity merely by seeking refuge in the Buddha or by mere faith in Him. It is not within the power of a Buddha to wash away the impurities of others

Every Buddhist tradition claims that the agent of a (karmic) act experiences its result, even though the two are not the same. There is continuity.

I can't say for every tradition, but right view, which is the 1st link on the noble 8-fold path of Theravada, is often said to include belief in this cause and effect. You can read about right view here.

More generally, there's no need for any belief to consider yourself a non practicing Buddhist. And I've read monks suggesting you practice zen without belief in karma.

There isn't one answer to this question.

For a start, there's more than one school of Buddhism: which have different doctrines, which you may find easier or harder to believe or understand.

Also,

It is said there are 84,000 dharma gates, or sometimes dharma doors, which is a fancy way of saying that people come to Buddhism in countless different ways.

One of the (many) variants of Buddhism in the West is apparently called Secular Buddhism, i.e.:

Secular Buddhists interpret the teachings of the Buddha and the Buddhist texts in a rationalist and often evidentialist manner, considering the historical and cultural contexts of the times in which the Buddha lived and the various suttas, sutras and tantras were written.

Within the framework of secular Buddhism, Buddhist doctrine may be stripped of any unspecified combination of various traditional beliefs that could be considered superstitious, or that can't be tested through empirical research, namely: supernatural beings (such as devas, bodhisattvas, nāgas, pretas, Buddhas, etc.), merit and its transference, rebirth, Buddhist cosmology (including the existence of pure lands and hells), etc.

Or, re. the traditional schools, the answers to What teachings do all schools of Buddhism share? list many doctrines, none of which actually includes "faith in hell and heaven, Karma, rebirth, other realms". Maybe they were omitted by mistake, maybe they're not a doctrine that all schools share, or maybe they're just not important enough to mention.

IOW I think you can "become a buddhist" without much belief in, without faith in, and/or without much understanding of these doctrines.

I guess my introduction to Buddhism was the four noble truths -- and I'm not sure that took any faith at all. The first two truths seem self-evident to me (or matches my experience), the third would stand to reason as a logical corollary of the second, and the fourth therefore seems worth investigating further.

I agree with your question though. In my first question on this site I commented that a doctrine of "rebirth" might be "a reason why some people might reject the Buddhist teachings entirely without close examination".

For the sake of answering this question more completely, you might want to read more about what "faith" means in Buddhism. For example, according to one school (quoting from The Jewel Ornament Of Liberation):

In that case, what does “faith” mean? There are three kinds of faith: trusting, longing, and clear.

III. TRUSTING FAITH

Understand that this faith depends on the topic “cause and result”—the Truth of Suffering which comes from the Truth of Causation. Furthermore, it comes from trusting that happiness in the desire world is the fruit of virtuous causes. Trust that the suffering of the desire world is the result of nonvirtuous action. Trust that the happiness of the two higher realms is the result of unshakable causes. Trust that by engaging in the nonvirtuous actions of body, speech, and afflicting emotions, which are called the Truth of Causation, one obtains the five afflicted skandas, which are called the Truth of Suffering

IV. LONGING FAITH

Understanding the extraordinary nature of unsurpassable enlightenment, one follows the path with respect and reverence in order to obtain it.

V. CLEAR FAITH

Clear faith arises in one’s mind by depending on the Three Jewels. Develop devotion for and interest in the Buddha as the teacher who shows the path, the Dharma which becomes the path, and the Sangha which guides one in order to accomplish the path.

To the extent that there is faith, it maybe doesn't mean "belief in phenomena which can't be observed." Even if you can't observe "the two higher realms", you can observe the "afflicted skandhas" (or conversely, something like "lack of remorse" after virtuous behaviour).

Admittedly that's expecting to observe phenomena such as happiness, desire, virtue, and affliction -- maybe it's difficult to observe those "scientifically" (though, speaking of "scientific observation", people have begun to use an MRI to test monks' ability to use meditation to affect their own happiness), but it is a kind of thing you can evaluate for yourself.

One of Buddhism's own summaries of Buddhism (i.e. Buddhist Dhamma) is that the teaching doesn't require a lot of faith -- i.e. it doesn't say about itself, "Buddhism is ineffable, mysterious, the chief virtue is faith, and it can only be understood through faith." Instead here is an example of what it does say:

The Dhamma is well declared by the Blessed One: visible here and now (a.k.a. actual, belonging to this life), immediate (or maybe timeless), inviting to come and see, effective, to be individually (by oneself) ascertained (known) by the wise (intelligent).