From Godfather:
The cat Vito/Brando holds in his first scenes in the study on the day of his daughter’s wedding was not scripted but a stray who wandered onto the set and who Brando picked up just before shooting began.
Luka Brasi/Lenny Montana’s bungled " Don Corleone, I am honored and grateful that you have invited me to your daughter… 's wedding… on the day of your daughter’s wedding. And I hope their first child be a masculine child. I pledge my ever-ending loyalty" spiel, while not exactly ad libbed, was natural. Montana was terrible with memorizing lines (by some sources the scene of him rehearsing his lines at the wedding reception is really the actor rehearsing his lines) and Coppola told him “we’re going to do this in one take, if you make a mistake pretend you’re really Luka and just keep going…”.
Vito picking at his grandson Tony with the orange skin/monster teeth in his mouth, while it may not have been the first take, began as an ad lib. It was a trick he used to play with his own kids. Giving the kid the insecticide canister was also an ad lib- it happened to be on the set for the plants and adds to how natural the scene works.
Many of the close-ups from the wedding reception are ad libs. Except for the parts that was necessary for the plot (e.g. Sonny with the photographers, introducing Pauly) Coppola basically had a huge wedding reception and told everybody to have a good time while his camera crew went through and filmed people at random singing, dancing, eating, and enjoying themselves. Tessio/Vigoda dancing with the little girl, Clemenza’s comment “What are you a dance judge?”, and other moments at the reception were improv.
I think- though I may be wrong- that Caan’s biting of his brother-in-law’s hands was an afterthought in the scene where Sonny whoops ass on Carlo. The actor who played Carlo said he was black and blue after filming that scene for two days. (There’s also a famous blooper in that scene when you hear a punch that never connects to Carlo’s face.)