re: Unscripted film shots or accidents that were kept in the final film (other than improv)

The cat on Brando's lap in the opening scene of The Godfather was a stray roaming around on the set and Brando picked it up and put it on his lap just before the scene started. Coppola liked it so much he left it in but they had to redo the audio because the cat was purring too loudly.