Yay, a thread about playing with phones! I freely admit to having spent way to much time sorting out my android experience. For a while there, it was like a hobby.
I use the powertoggles widget daily because my phone has poor settings access. Having instant homescreen toggles for wifi, flashlight, gps, and battery performance is handy, and not available naturally in all versions of android.
I use a (paid) podcasting app called doggcatcher that automatically downloads all of my podcasts every night while I'm plugged in and on wifi, so no data usage to always have new podcasts on tap. It automatically deletes things I've listened to, and has lots of settings for what to download and store.
Past that, I probably use google maps and navigation more often than anything else. They're default apps on most phones now.
I use google voice for voice calls when at home since I'm on a limited minutes plan.
I use MightyText to sync text messages across all of my devices. It has a chrome extension that lets you send/receive texts from your browser, which is handy when I'm at a big screen and my little one beeps at me.
I use a white noise app called Relax and Sleep to generate background noise for a fussy baby when we're away from home. Also good for calming nervous dogs on holidays with fireworks.
For news aggregation, I use Pulse. It pulls content from all of my specified sources and consolidates it one easy place, without ads, in a format that's easy to read and navigate. And it circumvents paywalls for things like the NY Times.
I use APW widgets for bookmarks and my google calendar. I like the interface on the calendar better than any other gmail calendar app I've used.
I've been playing with the new Yahoo Weather app, which has recently had a lot of press because it's included in the new iphone OS. It's kind of nifty, it will scrape the web for pictures of your current location with the same weather as you currently have, so, it's kind of like having a widget that serves the same purpose as looking out a window. But of course supplemented with weather data like temp and forecasts.
And because I have a really old and slow android phone (optimus elite), I've replaced my stock (Virgin Mobile) launcher with Holo Launcher. It's been customized pretty heavily with monochrome white icons on a dark background, SiMiClock for time on the homescreen, and notification apps that update my email/text/voice icons with little numbers overlaid on the icon, which I find very handy for keeping easy visual track of what I have waiting to deal with.
With some minor tweaking, this Holo Launcher setup is quick and buttery smooth, much nicer than my old Evo of my wife's HTC One. Newer phones tend to be so heavily skinned that those fancy fast processors don't give you much improved user experience because they use up the cycles on fancy graphics. My slow phone doesn't have 3D animations but it's far more pleasant to use because it's so smooth, and as a side effect I get great battery life (like three days) because it's not spending any power on useless prettiness.