Oh, man. I deal with a lot of anxiety both online and offline so mostly it's multiplayer games for me.
League of Legends is something I was really into for a while (kind of getting back into it now) but I didn't actually play that much of it because I physically couldn't bring myself to queue up. I could sometimes do it when playing with some friends but to this day I've never solo queued.
Fighting games were another. Last generation I bought SSFIV, MK9, and Injustice but never played any online for similar reasons. Just way too nervous and never got any good at them.
Most recently is Destiny 2. With the original Destiny I managed to scrounge up enough friends at the time to do a run at the Vault of Glass (never beat it though) but with Destiny 2 I don't have enough friends to do that so I went with the Giant Bomb LFG group. I found some cool duders to raid with twice (still never beat it though) but my anxiety just gets to me and it becomes really hard to think about even attempting to do it.