I like topics like this. A few I have that... might not be so well known, what they do, and a link to get them. In no particular order.
Smart Audiobook Player. One good way to pass a lot of time when you've nothing better to do is have a story read to you by a talented voice actor. Audiobook programs are less common than eBook readers, and of course far less common than music players. You could use a music player to play an audiobook, but you probably shouldn't. Audiobook players have extra features that set them apart. Like for example, if you pause Poweramp, you want it to resume when you come back. If you pause this, when you come back, it'll have reversed up to 30 seconds, depending on how long you were gone. So you backtrack a little. This app is free, with a $2 IAP for pro features, which are free to use for the first 30 days.
TuneIn. This one's free. I heard the paid version went up to ten bucks. Couldn't believe it. I bought it when it was one dollar. All it adds is recording. Free version is fine. Might have ads, I don't know. Anyway, I'm a transplant. Born in California, residing in North Carolina. Wife is from Connecticut. TuneIn, which you can also use on the web on your desktop or notebook computer, lets you listen to radio stations from all around the world. So I can listen to hometown stations, such as they are (our rock station went belly up... classic rock station is still great though). Or you can listen to stations from other countries! Call me old school, but I prefer this to Spotify and Pandora. The streaming itself is free, but data rates etc.
Perfect Screenshot. Not a screenshot app. It's a device frame generator. Puts your screenshots into a frame of your phone, so you can post a picture of your phone displaying the screenshot, not just the screenshot. Helps if you're a developer. Anyway, there are a few apps like this. All in development hell. They might have the last couple Nexus phones, but nothing recent outside of that. Perfect Screenshot has modern devices, and is actively developed. They just put out an update that has no device frames except Nexus 5, and you have to download the frames you want individually. This makes it smaller, and allows the developer to easily add whatever phone people ask for. Completely free.
Markers. It's a painting program, that can load pictures for you to draw on. Probably best with a Note device, but you can use your fingers, and it does multitouch. I don't use it often, but it's a pretty cool tool to have in your kit. Also, I took a picture of my niece, drew a crown on her head and a magic wand in her hand (she was about five at the time), and we got it printed. Far as I know her parents still have it on their fridge. Completely free.
SeriesGuide. If you watch shows, live/DVR, Netflix/streaming, or home video, you might want to keep track of what episode you're on. Maybe you want to rate episodes 1-10 as well? Then you want a Trakt.tv account, and an app to manage it. This is the best app I've found of its kind, and I've looked. Without Trakt it still works, but strictly offline. Much better to go to Trakt, set up an account, and add all your shows, and your progress with them, on the site, and then sync to the app. It can also tell you when shows are about to come on, via notification or widget. Also logs movies and anime (shows and movies). Completely free, but a paid subscription (recurring or one time) adds features you probably don't need (mostly, it's to support development).
Pocket MAL. Basically the same thing as SeriesGuide, but for MAL (My Anime List). This is strictly for anime. There's nothing wrong with using Trakt for anime, and in fact Trakt is better as it has names in English and Japanese — MAL insists on Japanese. But if you want to talk anime on the Net, MAL is the tracker of choice, and fellow anime fans can compare your MAL to theirs and get a compatibility rating. So, kind of a necessity, but searching can be challenging. Completely free.
Our Groceries. If you buy groceries, you will want this app. There may be better grocery list apps (in fact, I'm sure there are) but what this does, is syncs your list with whoever you want. Roommates, significant other, spouse, whatever. Now Google Keep does that too (just recently) but isn't optimized for shopping lists. Free with ads, or five bucks to banish them. So my wife and I paid ten bucks. But we made it back. Here's what we did. Her at Walmart with a wired headset, me at Food Lion with a Bluetooth. One list, and we'd both go to each item. Whoever can get it cheaper puts it in the cart. You only have to do that a few times to cover the cost. Most stores raise the price of X to subsidize the sale on Y. If you divide and conquer across competitors, you'll save a lot.
Walmart. I feel dirty for posting this, but hey, free money is free money, right? Expanding on what I was just saying about saving money? If you shop at Walmart and make an account on their site, you can log in with that account in the app. Scan your receipt, and if they find anything you bought cheaper elsewhere, they'll refund you the difference. It builds up until you pay out, at which point your phone becomes a scannable gift card. The credit doesn't expire. We're up to 12 bucks now, my wife told me. Oh and if we happen to find a receipt... damn right we're scanning it! I'm not digging through trash cans or picking up receipts in the parking lot, but say if one were laying around. I scanned a coworker's receipt I know doesn't use the app. This is totally free, as in the app, and the service.
I have a bunch more apps I use every day, obviously, but I'm just highlighting the less popular ones you probably haven't heard of. None of them were mentioned at least on this last page. If they were mentioned before, kudos to that user, you found a less popular gem (well except Walmart). Then there's the contested apps. Music player, launcher, etc. For those I think it's best to find your own way.