My all time favorite, Robotech

Especially the New Generation series

While it isn't 100% war and conspiracy, Attack on Titan definitely has elements from both of those within it.

I highly recommend Attack on Titan if you like intense shows that don't bullshit you with "no one ever dies" and the hero is always there on time.

Hunter x Hunter is also recommended, though it is more similar to action anime with the whole mystic powers and stuff. It looks childish from the first episode (the 2011 reboot, at least as it's the only version I have watched), but it gets dark pretty quick.

Space Brothers is also good, but it's very different than most anime. It isn't filled with fan service, isn't quite drawn the way a lot of anime are, and is very much set in reality. Its main draw would be the character development rather than the story itself. It has many funny moments.

Gintama is hard to recommend to someone new to anime, but I do recommend it after watching some of the other bigger anime titles as it often references them. Gintama is a comedy anime with occasional serious action moments. Not for everyone and has a lot of dirty humor littered all over, so it really depends on your type of humor whether or not you'll like it.

If you have been living under a rock, start with Ninja Scroll & Ninja Scroll 2. You can also try out the series collection to see if you like, but I wouldn't recommend it. Ghost In The Shell & GITS 2 are good. I don't know if there is a Ghost In The Shell 3. The Sky Crawlers if you want a story but it drains me. You should also try out "Blue Drop".

Clannad! you have to watch it first!!! the greatest anime ever

then afterwards go

Kanon 2006 (9/10)
Full Moon Wo Sagashi (10/10)
Sword Art Online (8.9/10)
Air (8.6/10)
Angel Beats (8.6/10)
Pokemon (9.6/10)
toradora (9.3/10)
The familar of zero (9.1/10)
Inuyasha (9.7/10)

The score next to the anime list are what i rated them :P

COWBOY BEBOP.

You say you like war and conspiracy, huh? Fullmetal Alchemist or almost anything with Gundam in the title would suit you well.

I keep a rotation of about four different anime series that I watch at a time: two super long runners that never seem to end, like Naruto and Bleach (DBZ, InuYasha, and One Piece are also good for this). These kinds of shows don't take a lot of brain power and allow you to either watch sparingly and come back, yet pick up quickly on the plot, or to go on a binge and watch a dozen episodes a day, if that's your thing.

I also watch two series that have a designated beginning, middle and end based on the 13/26 episode season format, like Ghost in the Shell, Blood+ and Death Note. These kind of shows are usually better in terms of plot and character development, because they contain far less filler and waste less time getting to the point.

Hulu is a great place to find lots of free anime. PLEASE support legal+free anime!

This. I've converted many, many people into anime lovers with that show. EVERYONE loves Cowboy Bebop.

I feel a little sick to my stomach that no one mentioned Akira or Serial Experiment Lain. Classics!

SERIOUSLY people STOP recommending Clannad. You're gonna put this guy off anime before he even begins. Clannad is an amazing anime, but it's not an entry level anime. Recommend him stuff that is good for someone who is just starting to watch anime, not ones which are enjoyable for seasoned anime watchers. Plus it's a very slow slice of life/romance anime that you really have to sit through to enjoy since it takes time to build up. Afterstory though, I almost cried at how amazing it gets in the second half.

I would personally recommend Naruto, Attack On Titan, Full Metal Alchemist, Bleach, Death Note as anime to get your feet wet. They will get you used to anime in general. After that it really depends on your taste. There are a LOT of anime and you just have to find the types you like best. Those can be separate recommendations in and of themselves.

I will however recommend two crazy awesome anime. GTO (Great Teacher Onizuka) and Level E. First one is one of the funniest anime ever made and the second one is just a straight up troll anime. It's purpose is to troll the audience.

Ouran High School Host Club
K-ON!
Death Note

All the ones I've watched so far, and they're all brilliant - for the record, I'm looking for a new one; does anyone have suggestions for me?