I'm extremely uncool and don't have a phone, but I do have an old-as-hell tablet that I occasionally use apps on, so I guess that's kind of the same thing. You know. A little.
A lot of my apps are specific to things I do personally (like when I played Rift - the MMO - I had the mobile app and checked it all the time, did the scratch cards it offered so I could get ingame rewards every hour... those were dark times, lemme tell ya) but there are the basics: gmail, skype, a browser of your choice (or you could just use the built-in one, w/e works for you)...
I'd say tumblr, because I DO have that app, but dear lord it's terrible. I used it only for my RP accounts so I could check and see if someone was pinging me for a thread or leaving me messages while I was afk. I honestly wouldn't recommend it unless you're super bored and super tumblr addicted.
Other things that relieve boredom though: the standard Bejeweled and its 200000 knockoff games, the Michiri Neko stuff (which is so cute you'll freaking DIE I swear - Michiri Neko Mix lets you "make cats" by throwing different kinds in a jar and it sounds stupid and you know, okay, it IS stupid, but it's adorable so idgaf) or other qt animated cat things (how obvious is it that I'm a cat person?) and uh... I think people were playing Lineplay and stuff a while back but I haven't tried it myself. Maybe consider getting an emulator app if you want to game on the go.
Productivity apps like evernote and dropbox are always good to have if you need to access files and stuff while you're not home, too.
(Touch Detective) Neo Mushroom Garden, Metal Slug Defence and Rhythm Of Fighters.
I really liked the characters in the Touch Detective NDS games so it's kind of fun to see them again.
MSD and ROF reminds me of playing the games on PS2. SNK have made ports of the games, but it's not the same without buttons (and it plays pretty badly). I think it's cool that they have relatively free spinoff games that really work well on mobile.