Essentials, let's see
Screebl: keeps the screen on when the phone is held in a usable position (configurable). Allows me to set the screen timeout to the minimum (15 seconds) so that when I set my phone down, it shuts off quickly, but the screen stays on if I am reading an online article. Also has an optional stillness timeout so that, if it is still within screen-on range while on the table (maybe gets set on something that holds the phone at an angle), it will still timeout after a period of time.
Lookout Ad Network Detector: shows me which apps push those irritating notification bar ads.
ShopSavvy: scans the barcode of most products to give comparison prices both online and nearby.
Bible (YouVersion): Bible app with downloadable translations, as well as audio, and reading plans.
Any mobile banking app: check accounts, make transfers, pay bills, some even allow you to deposit checks from your phone.
Google Drive: cloud storage, as well as editable documents in the cloud.
ES File Explorer (or any good file explorer app): allows you to access files on your sd card without plugging into a computer. Good for moving stuff into specific folders, cleaning out your card.
Evernote: note taking app that syncs with your computer.
Movies by Flixster (or Fandango): check movie times, reviews, view previews, buy tickets.
GroupMe: group chatting app. Allows me to chat with my buddies all at once, rather than trying to text all of them at once.
Handcent: text messaging app. Much more configurable than stock app, will even allow to set different tones for different contacts for texting.
JuiceDefender: turns off mobile data with the screen, with occasional connection to allow for syncing of email, calendar, etc. Saves much battery.
Netflix: movies on my phone? What?!?!
Simple Calendar Widget: nice looking calendar widget so I can see everything from my homescreen.
BattStatt: battery percentage widget in a text style
SoundHound: music ID app. What's that song? SoundHound will tell you. You can even hum or sing it, and it can ID it for you.
Starbucks: cause I'm an addict and I can never keep track of my cards. You can pay with your phone at most Starbucks locations, as well as add new cards and add value to your current one.
Zedge: wallpapers (good quality) and ringtones.
Angry Birds (all of them): I know, how hipster of me, but they're fun.
Armored Strike: turn based tank warfare with online play. Honestly, I don't play this one much anymore, but I do find it fun and come back to it every once in a while.
Flick Golf! (and Extreme): name pretty much says it. Fun game.
Fruit Ninja: fruit slicing game.
Homerun Battle 3D: homerun derby game.
Road Warrior: post-apocalyptic racing game. Upgradable weapons, vehicles, armor, vehicle customization. Do flips for a boost, with online play, my current favorite game that gets me in lots of trouble with my wife.
Some of these are paid, some are free, all are available in the Play Store, and all of them go back on my phone immediately if I have to do a reset or get a new phone. Check them out, use trial versions (or free versions, which may not include all features, but give you a good idea of the app), and have fun. It's amazing what these phones can do.