Thus far we have come up with Crystal the Monkey at 24 credits.
Astathe dog from the 30s and 40s with 21 credits.
Roy Rogers’s horse Trigger has 96.
The original Rin Tin Tin has 28 and was one of the biggest stars in Hollywood in his heyday.
Trigger is a very good one. Probably the answer.
Zamba racked up 27 credits, including the title role in The Lion (1962). A very impressive career and a remarkable animal.
Mr. Ed appeared in 145 episodes of his TV show, 1961-66.
I was surprised that Rin Tin Tin, Lassie, and Topper (Hopalong Cassidy’s horse) did not have more credits.
I was also surprised that Rin Tin Tin, Rin Tin Tin Jr, Rin Tin Tin III, and Prince the Dog each had a separate entry.
Cheetah the chimpanzee only has two credits, though, and is not credited for most of the Weissmuller Tarzan movies. So I think imdb’s records are incomplete.
Bart the Bear has about 20. He should have gotten an Oscar nod for his role in The Edge. 
Higgins (who played Benji) only has 7.
Pete the Dog (a/k/a Pal) has 95 credits, almost all of them from early entries in Hal Roach’s Our Gang/Little Rascals series of shorts. IMDb has a separate entry for Pete the Pup with 26 credits, mostly in later entries of the same series. They are presumably two different animals, but I’m no expert.
He’s nowhere near the record, but honorable mention to Terry (III) with 21 credits. You know him as Toto in The Wizard of Oz.
Mister Ed was his stage name. His real name was Bamboo Harvester. But the studio told him that was too ethnic.
I would have thought Cheetah would have more credits. But reading the Wikipedia article, apparently there were multiple chimps who played the role in various films.
If you count individual episodes of a TV series, Lassie appears to have 365 credits for his self-titled show. But it’s a lie. Lassie was portrayed by six different dogs (all male) during the nineteen seasons the show ran; Pal (the movie Lassie), Lassie Junior (Pal’s son), then four of Lassie Junior’s sons, Spook, Baby, Mire, and Hey Hey.
Moose. The real star of Frasier.
In the show didn’t Mr Ed tell that guy what his name is, regardless of his human given name?
Clearly SAG/AFTRA need to address the failure of the studios to properly credit these actors.
No, you need WAG/ARFTA for that.
And I’ve always thought it odd that a dog named a synonym for “girl” would not only be male, but of the one dog breed where males and females are instantly distinguishable. Female collies don’t have that thick mane.
As a six year old watching Lassie on a huge black & white TV set, I had no problem with a transgendered or possibly-just-cross-dressing gender-bending dog.
Oh, and the high point of that year was that I got to meet King The Wonder Dog (faithful companion of… Sgt. Preston of the Yukon!"… and too much a character actor for this list).
If we count individual episodes of a TV show (I didn’t), Trigger adds another hundred to stay on top.
I have no idea as to Lassie’s gender identity, but if e were cross-dressing, e would have trimmed down the mane.
This is a strong joke that deserves respect.
Also, off-topic — “e”? Is that like “s/he,” but more elegant and au courant? I like it!