One that comes to mind is Babai's spectacular algorithm for [Graph Isomorphism][1]. He chipped away at the problem for over 40 years (while doing much other work of great note). He started in 1977 (according to one popular article). 

In 1983 He and Luks came up with an algorithm with worst case run time $2^{O(\sqrt n \log n)}.$ That was the state of the art for over 30 years. 

In  November 2015, Babai announced an algorithm  with worst case running time $2^{O((\log n)^{c})}$

Like Wiles proof, it had an error which was later fixed. I don't know that it is universally accepted, but 


I think most in the field were pretty amazed by the result. Even though his plan of attack was known and he published results along the way, a result that good was thought out of reach for a decade, if at all. 

I didn't find a good quote by him but I've heard talks where he discusses the result and his wrong turns and breakthroughs over the years. There are many talks by Babai on the web, [here is one][2]. If it is like talks I've heard, he discusses (someplace in those 90 minutes)  the process. 

  [1]: https://en.wikipedia.org/wiki/Graph_isomorphism_problem
  [2]: https://www.youtube.com/watch?v=r-nCYbX_Au0