Here's my top 10 in no particular order... I've been out of touch with anime until just recently so getting back into it I felt like this thread was a great way to check out stuff other people liked and maybe give you guys some alternative anime you may not have checked out yet. I have a LOT more I want to include in my top 10 but I tried to avoid doubling up on anything that's already been said before.
1. Kino's Journey
One of the first shows I've watched, it really helped me through a tough time in life. The serene sense of peace and reflection this show invokes is reminscent of Mushishi for anyone who has seen it but there is something really special about the way Kino's Journey handles human ugliness and beauty. The TV show is flawed in a number of ways but its always going to be a special show to me.
2. Re: Creators
A more recent show, this was the show that got me back into the swing of things with anime. I really, really like this show, but it certainly isn't for everyone. As a creator/artist/writer, this show hit me in all the right areas, it was really something I could directly relate to and I think because of that I see something in it that some other people may not be able to see themselves. If you are an avid creator like myself, I highly recommend giving Re:Creators a chance. The characters in this show (for the most part) are all written excellently and the interactions are quite "real" to me. Also Hiroyuki Sawano soundtrack.
3. Keijo!!!
Ok, hear me out on this one. I'm not normally into shows like this because I'm a total hipster but I had to give a shout out to this little gem. Keijo is ridiculous, its stupid and borderline exploitative BUT there is actually something of substance here believe it or not. People give this show a really bad rep (for good reason) but I personally have not had as much fun with an anime as I did with Keijo. It's absurd, its ridiculous, it's completely aware of that and I think because of that it comes off as a really fun show to me. I recommend keeping an open mind when checking out Keijo and you might just find something in it you didn't expect.
4. K-ON
There isn't enough I can say about this show. In the west, I think K-ON has been unjustly labeled as just another "moe" show by some people. I can't stress how wrong they are. Your mileage will vary a lot with this show but it is one of the most popular and enduring anime shows in Japan for a very good reason. There is a lot going on in K-ON that may not be readily apparent but if you haven't seen K-ON before or given it a fair shot, I highly recommend coming into it with an open mind and see if it sucks you into it's world like it did to me.
5. Kill la Kill
I love Studio Trigger and particularly Imaishi's works. I could list every Trigger show here as a recommendation but I'm gonna stick with the one that made the biggest impact for me. Kill la Kill is a show that doesn't pull any punches. It has an identity, it has an ethos and it sticks with it whether or not you like it. I can appreciate that brazen attitude and youthful recklessness that this show oozes. It's emotional, it's smart, and it wears it's themes with pride. Once again, you will have to keep an open mind about it but whether you're checking this show out for dumb fun or for something emotional and a little deeper, you'll find it all here. Can't recommend this one enough.
6. Kara no Kyoukai
Once again, like Studio Trigger, UFOTable is my other favorite animation studio. They worked on the phenomenal Fate series, particularly Fate/Zero and Unlimited Blade Works but perhaps you haven't heard of one of their lesser known works. Kara no Kyoukai is confusing and feels like Kinoko Nasu wrote it's story entirely in a stream of consciousness style. There is something that is still incredibly interesting about the world and characters that he created, even if his writing isn't always on point. Kara no Kyoukai is also helped a lot by it's Yuki Kajiura sound track. There are 7 movies in total, so it might as well be an anime
7.Shimoneta
Legit one of the funniest shows I've ever seen. It's super smart despite it's subject matter and appearance and this show literally only exists to criticize one of the most confusing aspects of Japanese culture. Shimoneta is an insanely good time, especially if you are familiar at all with Japanese culture. Shimoneta is raunchy, hilarious, and is addressing a very important issue for Japan going into the future, even if it's only doing it for comedy.
8. Nichijou
I love this show. I love every part of it. It is the 4koma absurdist humor done in the best way imaginable. The animation is top notch, the characters are endearing, the skits are hilarious. I have literally nothing bad to say about this show. Whenever I feel down, or I've had a bad day, I've been known to just kick back and watch and episode of Nichijou. It has never failed to pull me out of a slump.
9. Jormungand/Black Lagoon
I'm putting these shows together because they are similar (but very different as well). Black Lagoon was one of the first anime I've ever watched, so despite it's issues its always going to be one of my favorite shows. I have yet to see a character as bad ass as Balalaika. Jormungand is in a similar vein but it's a little bit more fun of a show.
10. Danna ga Nani wo Ittieru ka Wakaranei Ken
This one is up here only because my relationship with my wife is like this. We both loved this show and it hit home for us on more than one occasion. It's one of those anime shorts so it isn't a full show or anything but I loved it.