Just thinking aloud here: obviously most series of cartoons, and series of comedy films in general, don't have much continuity. But some film series go further and have the same characters, under the same names, who not only don't remember anything that happened in the previous instalment, but don't even remember each other -- that is, they literally meet for the first time in every film.
The Warner Brothers cartoons did this a lot, especially after the war when there were more series with more than one continuing character. In the Warner Brothers universe, every cartoon is the first time Yosemite Sam meets Bugs Bunny, the first time Sylvester sees a baby kangaroo and mistakes it for a giant mouse, the first time Pepe Le Pew sees une belle femme skunk. Sometimes Sylvester knows Tweety and Speedy already, but sometimes he doesn't. Same with Elmer Fudd and Bugs Bunny; there are a few cartoons that kind of imply that they've been through this before (like "Easter Yeggs" where Bugs clearly recognizes Elmer the first time he sees him), but more often they're just meeting for the first time.
I don't think other studios went quite as far in wiping the slate clean, but I seem to recall some Tom and Jerrys where they don't know each other, some Woody Woodpeckers where someone from a previous cartoon is a complete stranger to him. And in general, most cartoons would not have characters talk about meeting each other in a previous instalment or act like they have any kind of "continuing" history with each other. A possible exception would be some shorts from Disney, who wanted Mickey, Donald et al to have friends and family. Hence cartoons like "Mickey's Birthday."
Television cartoons don't do this as a rule. Maybe some of the early ones did, but usually you get two or more characters who have been together before the episode started, fighting villains who are already more or less known to them (like Boris and Natasha). Even shows where the characters can appear at any point in history will still not go as far as to show them not remembering people they've already met; it just isn't done. (Unless this is something like Mr. Burns not remembering Homer, which is different.) So Animaniacs characters could have stories that took place any time since the dawn of time, but once they met Dr. Scratchansniff, they knew him and didn't keep on meeting him in subsequent cartoons. Ren and Stimpy usually knew each other. The closest thing to an old-school cartoon device on TV is killing Kenny on South Park, and they've mostly dropped that.
So: what are some other cartoon series, or other film series, that re-set to zero like that, where not only are there no carry-overs from instalment to instalment, but the characters actually meet each other again for the first time, over and over again, for eternity? Which is kind of depressing if you think about it too much.