In 1978 Roger Apéry proved the irrationality of $\zeta(3)$, giving a talk named "Sur l'irrationalité de $\zeta(3)$" which is known for being unusual. There don't seem to be many accounts of the talk, but from some claims that do exist, while his claims were unbelievable, his presentation was quite droll, and it wasn't until after the talk that it was seriously considered that he was correct. (1)
A possible recent second example is the proof that for every positive integer $n$, there exists a pattern in Conway's Game of Life which oscillates after exactly $n$ generations (since $n$ is called the "period" of such a pattern, this conjecture is known as omniperiodicity.) By 1996 it was known (2) that an oscillator exists for every period $n\geq 61$, by constructing an infinite family of similar patterns based on shuttling active regions around various kinds of looping tracks. This left a list (2) of eighteen periods less than $61$ with no known oscillators - the website LifeWiki gives a short timeline of the last twelve periods to be discovered, and some witnessing patterns. (3)
The final construction was of an oscillator with period $41$, which was first posted without much fanfare into a LifeWiki-associated Discord chat room by Nico Brown in July 2023. It was soon after copied over to the the ConwayLife.com Forums by the same author. The resolution was soon after catalogued on Adam P. Goucher's blog Complex Projective 4-Space in the post "Miscellaneous discoveries".
Edit December 2023: The paper "Conway's Game of Life is Omniperiodic" now posted on arXiv contains more details about the chronology of the result and the constructions.
(1): A. van der Poorten, "A Proof that Euler Missed ...", The Mathematical Intelligencer vol. 1, 1978.
(2): D. Buckingham, "My Experience with B-heptominos in Oscillators", October 1996. Hosted on conwaylife.com, accessed 25 October 2023.
(3): LifeWiki, "Omniperiodic". Wiki article, accessed 25 October 2023.