Since no one mentioned it yet, I recommend Sword Art Online ...

Parasyte
Akame ga Kill
Attack on Titan
Sword Art Online
Shingeki no Bahamut
Owari no Seraph
Gangsta
Tokyo Ghoul
The Irregular at Magic Highschool
World Trigger

Depends what genras you like and if you don't mind fan-subbing or do you want them dubbed?

You could check out some of the manga sites first and you can be fairly certain that if a manga is popular as an example over 40 chapters then there will be an anime made for it.

I do agree with "Riverstyx" those anime are great however they are from our own personal likes.

Personally I did like "Log Horizon" over Sword Art Online" although that should not detract you from watching them both. If you like fansubbed then for Log Horizone "Commie" (mkv) or "Deadfish" (mp4) are pretty good. For SAO try fansub group "Hiryuu" (mkv) or "Deadfish" (mp4).

If you liked Dragon Ball then you will also like "One Punch Man".

For people who liked "One Piece" which is still on-going you may like "Fairy Tail" (265 episodes and counting). You can get about 100 episodes dubbed in English but I personally prefered the fan-subbed versions since I found that the Japanese voice actors voices really did fit the characters. I can understand a few Japanese words but I am definately not fluent in the language and cannot read the characters.

One anime which is fairly recent is "Gate" which is just started it's second season. This anime does follow the manga but the manga is very much more graphic in terms of fighting compared to the more sanitised anime although that should not detract you from watching it.

I do reccomend downloading at least three episodes of any anime that takes your fancy and watching it before making a decision to continue.

Try "www.nyaa.eu" or "www.nyaa.se" for anime searches (no anoying adds) although there are other sites.

There have been quite a few episodes of monogatari since monogatari second, what about starting there ?
Also Oregairu is a great take on the school love comedy that peaked around 2012-2013.
Gundam IBO is the latest new universe in the gundam series and takes us back in the eighties with it's artstyle while being most inspired by 00's narrative. Definitely the best Gundam series since 00, and it's written by Okada.

Hajime No Ippo
HunterXHunter
Shingeki No Kyojin (Attack on Titan)
Sword Art Online (S1 and S2)
Akame ga Kill
One Punch Man
Steins Gate

Other Anime I've watched but I wouldn't outright recommend it to everybody (still good anime though):

Log Horizon
Noragami (S2 is called Noragami Aragoto)
Ao no Exorcist
Fairy Tale
Fate Stay Night
Kill la Kill

Katanagatari is really good.
Bacanno! is really good.
Eden of the East is really good.

You can binge One Punch Man now.

Hokuto no Ken!

One Punch Man (Good all round)
Shinsekai Yori (Amazing story that will you leave you jaw on the floor at the end)
Bokurano (Another good psychological anime)
Magi (Good fights, good story)
Noragami (Good story, good fights, one of the only truly funny anime)

There are others too but those are the ones I can think of right now.

Perfect Blue