If you type the phrase 'best based on true story movies' in a search engine you will find lots of lists, all different, and of course like all such questions it depends on your taste. However, there are certain titles which keep coming up.

The most interesting list I've found is from the IMDB. Their top film is 127 Hours, a movie chosen by many viewers for its shockingly real depiction of events. More classic films with a real-life basis include Zorba the Greek, which is based on author Kazantzakis's friendship with a Greek miner, and the very moving (though historically not very accurate) film about the life of US mathematician John Nash, A Beautiful Mind.

Another interesting choice is Alexander - the IMBD critics rate it highly, but it's had mixed reviews from audiences (at nearly 3 hours long, it's too much for some).
One new real-life based film you may like is The King's Speech - again there are historical inaccuracies, but it's a very good film.