The One V with Sense has a lot of built in apps that remove the need to download quite of a few (not taking, clock widgets, Calender widgets), etc. but here are some Android apps that I use a fair bit.
Games
-Words with Friends (Scrabble online)
-Draw Something (Kinda like pictionary)
-Cut the Rope
-Game Dev Story (only pay app I'll probably have on here)
-Home Run Battle 3D/Home Run Battle 2
-Mega Jump
-7 Words
Gameboid - may not be in the app store anymore, but who doesn't want to play GBA games on their phone?
*Note: I don't play a ton of games on my phone, these are just ones I've enjoyed at some point.
Productivity
-Astrid (much like Scott's mentioned Any.do. Not sure which is better, but a task manager app that can link with Google tasks and has a widget. I just noticed that HTC has a native app, so maybe use that)
-ColorNote. Not sure if this is redundant with HTC's Notes app, but a simple note taking app where you can post "notes" as widgets - like post-it-notes
-Dropbox. Use it. Also, you may have gotten some free storage with your one series phone. Use it.
-Evernote. I know this is integrated into HTC's Notes app, but haven't played around with it. I've been using the standard app for years. Take any note, picture, voice note and it will upload to the server so you can access it wherever. It's been handy a ton of times.
-Habit Streak. Like a task app but for reocurring tasks you want to do daily. Keeps track of how many days in a row you've done them.
-Realcalc scientific calculator. Maybe not neccesarily this one, but depending on what you can do, you can find a calculator with more usefull functions for yourself than the very simple stock one. From graphing to construction calculators, they'll have something handy in the Market.
Media
-MX Player - Best video player I've found.
-Quikpic - Best Photoviewer I've found. I use it out of habit, but HTC's Gallery app is pretty good as well.
-Poweramp - Great music player. Free trial and then pay.
-Winamp - Free
Skifta - If you have a media server/shared media stored on PCs, you can access it through this. You can also access it through HTC's media player, but I find that this is a bit faster in browsing the server.
Communication
Skype
WhatsApp - Multiplatform messaging service. Uses data. Might be the closest thing you find to BBM (Note, the second "checkmark" on a message doesn't mean that it's been read, just that the server has delivered the message to the recipient's device.
GoSMS - If you ever decide you want a change from the standard HTC messaging app, use this.
System
Astro - File Browser, etc.
Swype - Not in the appstore, but probably the best keyboard replacement out there. The HTC equivilent is close, but not quite. Have to sign up for beta on their site to download
News
-Reader. People like this. I'm torn, I find the web app just as good, maybe better.
-Flipboard Like Swype, you need to sign up for Beta on their webpage (for now). Pretty cool.
-Chive Everyone likes entertaining low-brow pictures
I'll add more as I come up with them, but there are a TON of apps out there for pretty much everything. Don't like the camera app? Get a new one, etc. If I was new to android, I'd check some Android news sites, they seem to have lists all the time.