There's a fair bit of annoyance possible with GIFs, a fact I exploited when Google+ decided to:
1. Allow animations in the "Hero" user profile image.
2. Make that image permanently fixed to the top of the screen on the profile page.
3. As users increased window size to see more text and less hero, the image would scale to as much as 3/4 of the screen.
4. The hero was included on the "hover card" for users and would show up in all kinds of locations throughout the service. I still have that collection....
I spent an hour or so researching "annoying gifs", and settled on a rapid red-green-blue stroboscopic rotation. I don't know the cycle time, but it was probably a few times per second.
Google eventually relented. Well, they eventually shitcanned the whole site, but that's another story.
Trying to find animated images that weren't mind-bendingly annoying to me is how I ended up with the "space alien cat" image and avatar. The original had a rotating set of astronomical images in the glasses, I eventually applied the Crab Nebula as a static fill (tried the Cat's Eye, but ironically, it didn't work particularly well). That's the hero I use at several sites (Ello, Mastodon), and is the inspiration to the avatar I use as well.
Later at another site at which gratuitous annoying gif use was prevalent, I found a few static-but-apparently-moving optical illusion images, again to make the point that annoying images are annoying. The point was somehow missed by that crowd. That site also died.
TL;DR: animated gifs and even static images can be plenty annoying.