According to non-inerrantist Christians, what are the strongest arguments against the inerrancy of the Protestant Bible?

The strongest argument are the actual problems within the text.

2 Samuel 24:1 states that God incited David to take a census of Isreal, but according to 1 Chronicles 21:1 Satan incited David. Both cannot be factually correct. There are other examples, but this is one of the clearest ones.

Other examples include things which we know are false due to advances in science, e.g., the earth resting on pillars. But inerrantists usually find sophisticated arguments to dismiss these.

Update: now that I think about it (and after I have witnessed a mod on this site delete comments under my question which revolved about this issue), I think the strongest argument against inerrantism to me, personally, is the attitude of those who hold it. The readiness to dismiss arguments which don't fit the narrative (see remaining comments below) and the lack of rigor (problematic verses? let's just define the problem away) of its proponents is not exactly helpful in making the theory seem intellectually reliable.

Okay I'll take the bait.

Depending on your point of view, the strong arguments would probably be categorised into two key areas, taking your evangelical notion of 'error' given in the question.

From within the Bible

Comparing parts of the bible that say different things, is considered a strong argument. It is also a very old view. I'm not going to research and link here to biblical contradictions, since they are widely available on the internet, but I might give a couple of examples:

First the death of Judas - compare Matthew 22:5, where Judas rejected the money and hanged himself, with Acts 1:18, where Judas used the money to buy a field, and then was disembowelled in the field. You can read an attempt to resolve this on Answers in Genesis which provides a fairly unconvincing explanation that ignores the differing accounts of what Judas did with the money and requires an implausible interpretation of Luke's choice of words in Acts.*

Second: the length of the flood in Genesis, is it 40 days and 40 nights (Gen 7:12) or is it 150 days (Gen 7:24).

I could go on, but I find this activity about as enjoyable as catching butterflies and sticking them to boards with a pin and it serves nobody.

From outside the Bible

Again this is information that is very widely available, since many commentators take a lot of glee in finding supposed errors in the bible. The basic idea is that knowledge carefully built up in various unrelated fields of knowledge comes together to refute biblical claims. I will only consider one example: the age of the humankind and of the earth (with the caveat that, as always, Christians who do not think it is literally true also do not think it is in error, but that it is metaphorical/poetic/mythical). In this case you have evidence from a wide array of fields of study with completely different methodologies, including:

Geology The study of rock formations allows a person to estimate the age of different strata, of different rock formations, and indeed of the whole earth. Understanding magma and plate tectonics allows you to track backwards to events in the far past - such as the creation of the himalayas by the collision of the Indian Subcontinent with Asia.

Biology and evolution Modern molecular biology dovetails extremely well with the fossil record. Basically you can look at two species and use mutation rates of genes and estimate when they had common ancestors, and by and large this fits very well with the fossil record of such ancestors actually existing. The reason this kind of thing is so pursuasive is that the molecular biologists are in their labs with the machines and computers and DNA strands, and the fossil hunters are out in the field with their shovels and brushes, and so the fact that there is an incredible convergence between the two is considered convincing.

Astrophysics Comes to the party with the agreement between the extremely old age of the Sun, from principles of nuclear fusion and star formation, and the age of the earth from geology. Also, the extreme age of the universe based on the expansion of the galaxies, led to predictions about the big bang and people theorised that a big bang would produce leftover background radiation, which was much later discovered to be pretty much exactly where and of the form it was expected

Archaology Particularly patterns of human migration, the age (40 - 60,000 years) of human settlements in Australia for example (Note that archaological evidence can be used to 'contradict' various historical claims in the old testament, I think some people would find it very convincing and others wouldn't).

Carbon Dating Fits nicely with archaological finds and geology to look at when living matter died (if in the last couple of tens of thousands of years).

Anthropology: In Australia, indigenous people have stories of changes in the environment that date back many thousands of years. For example, there are stories and archaological finds that date back to when Lake Mungo was filled with water, which was around 16000 years ago according to geology, and another indigenous story about palm trees being brought from the 'North' in the distant past matched with genetic analysis estimating that the local palm trees had diverged from Indonesian trees 30,000 years ago.

I think what Christian literalists sometimes fail to appreciate is that this isn't one field of 'science' having some uniform critique, it's various completely unrelated fields coming up with compatible conclusions through completely unrelated methods, that makes their arguments so strong. As Pope John Paul II said while speaking favourably about evolution (source, and original source in French (starting from para 4)),

New knowledge has led to the recognition of the theory of evolution as more than a hypothesis. It is indeed remarkable that this theory has been progressively accepted by researchers, following a series of discoveries in various fields of knowledge. The convergence, neither sought nor fabricated, of the results of work that was conducted independently is, in itself, a significant argument in favour of the theory.

*_{This is not related to your question, but to a Catholic or mainline Protestant these discrepancies are often seen as the result of the weaving together of different oral traditions that refer to the original events. In the same way that languages diverge, so do traditions and stories, and there are two divergent traditions on the death of Judas that Matthew and Luke are recording. So the fact that these depictions are different are not "errors". I want to answer this question in the spirit you've asked it, but I also want to be clear that not everybody shares the evangelical definition of, or obsession with, errors.}

The strongest arguments are from those who come from traditional sectors of the Christian faith. For example, recently, John C. Poirier did an examination of one of the key texts in formulating the doctrine of inerrancy. His book detailing the results of his linguistic study is, The Invention of the Inspired Text: Philological Windows on the Theopneustia of Scripture. The overview runs as follows:

John C. Poirier examines the “theopneustic” nature of the Scripture, as a response to the view that “inspiration” lies at the heart of most contemporary Christian theology. In contrast to the traditional rendering of the Greek word theopneustos as “God-inspired” in 2 Tim 3:16, Poirier argues that a close look at first- and second-century uses of theopneustos reveals that the traditional inspirationist understanding of the term did not arise until the time of Origen in the early third century CE, and that in every pre-Origen use of theopneustos the word instead means “life-giving.” ...

Than there is the Christian apologist, C.S. Lewis. He writes:

The earliest stratum of the Old Testament contains many truths in a form which I take to be legendary, or even mythical—hanging in the clouds, but gradually the truth condenses, becomes more and more historical. From things like Noah’s Ark or the sun standing still upon Ajalon, you come down to the court memoirs of King David. Finally you reach the New Testament and history reigns supreme, and the Truth is incarnate. And “incarnate” here is more than a metaphor. It is not an accidental resemblance that what, from the point of view of being, is stated in the form “God became Man,” should involve, from the point of view of human knowledge, the statement “Myth became Fact."

Although, to be fair, Lewis viewed his thoughts in a tentative manner. For example, he writes:

My present view — which is tentative and liable to any amount of correction — would be that just as, on the factual side, a long preparation culminates in God’s becoming incarnate as Man, so, on the documentary side, the truth first appears in mythical form and then by a long process of condensing or focusing finally becomes incarnate as History. This involves the belief that Myth in general is not merely misunderstood history . . . nor diabolical illusion . . . nor priestly lying . . . but, at its best, a real though unfocused gleam of divine truth falling on human imagination. (Miracles: A Preliminary Study [New York: Macmillan, 1947], 161n1, emphasis original).

That this was the view of his for the majority of his life can be seen in the 1959 letter that Lewis wrote to Clyde Kilby, a professor at Wheaton College, in which he states in part:

Whatever view we hold on the divine authority of Scripture must make room for the following facts...The apparent inconsistencies between the genealogies in Matt i and Luke iii: with the accounts of the death of Judas in Matt xxvii 5 and Acts i.18-19.

Of course harmonists, at least with the New Testament, are fine with agreeing with the various scholarly insights that have traditionally been offered to reconcile alleged contradictions. For example, legal scholar Simon Greenleaf, the author of "Treatise on the Law of Evidence" supports a harmonist understanding in his book "The Testimony of the Evangelists: Examined by the Rules of Evidence Administered in the Courts of Justice."

Lewis might very well have changed his viewpoint had he taken the time to more thoroughly examine the various positions.

The strongest argument for there being at least one error in the 66 books of the bible is that there are so many plausible candidates for such an error. Defenders of inerrancy are left having swat down hundreds of examples, using arguments from a simple copyist error (which doesn't count even though it became part of the textus receptus) to translation issues, mental contortionism to reconcile conflicting ideas, and arguing that a statement or entire story is poetic hyperbole rather than meaning what it actually says.

The evangelical scholar Mike Licona discusses this problem in a 2014 article here. For him, the problem with a strict adherence to inerrancy is that once a reader is convinced of even one real mistake by biblical writers, the whole house of cards falls.