My SeaMonkey (nee Netscape Communicator and then Mozilla Suite) allows me to set "Animated images should loop" (1) as many times as the image specifies (2) once (3) never. I have it set to once, so I've never been annoyed :-)
I'm not sure about other browsers as I've used SeaMonkey for web browsing and as an email client almost exclusively since 1994.
I've used SeaMonkey (AKA IceApe, nee Mozilla Suite nee Netscape Communicator) a lot in the past, but mainstream Firefox no longer has that menu entry.
Fiddling and Googling, the corresponding setting in about:config is image.animation_mode, which takes the values "normal", "once" or "none", but I suspect that it might also interfere in some way with completion of the page-load operation (the animation on the tab might take much longer than normal to settle down).
Regrettably, we live in an era where using about:config is quite simply beyond most people. Now obviously it's not beyond me, but the browser profile I use for simple documentation etc. already has as much scripting as possible disabled, and it's irritating to find a 90's style animation intruding.
MarkMLl