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In SN 7.21, Ven. Ananda requested the Buddha to visit the home of the lay brahmin Sangarava and teach him, to debunk his belief in water purification rituals. The Buddha consented. The brahmin neither asked to be taught, nor approached the Buddha for any other reason.
In SN 7.22, the Buddha approached the council hall of Khomadussa where householders and brahmins were having a meeting. They saw the Buddha coming from a distance and mocked him as being a fake ascetic, to which he responded by giving them a teaching.
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Putting the Pali qualifier to one side, without discrimination - if we ask ourselves what it means to teach?
Just by being - demonstrating the fact of the possibility of liberation - was one of Lord Buddha's greatest and most significant teachings. Providing an example of how to speak, how to teach, how to sit, how to walk - all of these were direct and immediate teachings.
The best way to teach a child is through example, not through words, right?
Did he talk without being asked to? Yes, often. Were some of those talks teachings? I would argue all of them were: Every moment of experience with a Buddha is going to be a very powerful learning experience. Every moment with a significantly realised teacher is a learning experience.
Did he give formal talks without being asked? I do not know.
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The Akkosa Sutta: Insult (SN 7.2) might be an example, perhaps a rare example. In it, the brahman wasn't explicitly asking for a teaching -- though he did go and initiate the conversation. It's interesting (or skilful) how the Buddha does it -- perhaps the brahman wasn't inclined at that moment to quietly listen to a discourse, instead the Buddha engages him in dialog using a rhetorical question.
He does give discourses to laypeople, villagers -- maybe often, the Kalama Sutta (AN 3.65) is one example -- I think the usual scenario in that case is that they respect him as a teacher. See for example the story of The Weaver's Daughter for an account of how he might often talk with laypeople.
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I think the question might need reframing. The Buddha's Dhamma is of two types (MN 117):
For me, the question is: "Did the Buddha ever offer supramundane teachings without being asked to?"
The account of the First Sermon in the Vinaya gives the impression the Buddha was pushy & offered to teach without being asked at a time when the five monks were not yet the Buddha's Sangha:
The Buddha said to the group of five monks, “Monks, don’t address the Buddha by name or as ‘friend’. Listen, I’m perfected and fully awakened. I have discovered the deathless. I will instruct you and teach you the Truth. When you practice as instructed, in this very life you will soon realize with your own insight the supreme goal of the spiritual life for which gentlemen rightly go forth into homelessness.
The Buddha was able to persuade the group of five monks. They then listened to the Buddha, paid careful attention, and applied their minds to understand.
In MN 49, the Buddha went to a Brahma-world to harass Baka the Brahmā, who did not attain any enlightenment from the Buddha's harassment. (However, my personal view is MN 49 is a late fake sutta).
As for mundane teachings, I expect a Buddha would provide these in the general course of meeting others socially.