Go straight off and watch
Tengen Toppa Gurren Lagann (aka
TTGL or just
Gurren Lagann). 26-ep series, or two movies. Best, watch the series first, and then at least the second movie which adds yet another layer of awesomeness to the final battles.
Also watch
FLCL (aka
Fooly-Cooly or
Furi-Kuri) - only 6 eps, but one of the craziest series ever, and the animation style feeds straight into
TTGL.
Actually, between them in development is another awesome 6-parter:
Diebuster (aka
Gunbuster 2 or
Reach for the Top 2) - but don't watch that until you've seen the original
Gunbuster (aka
Reach for the Top), because although the stories are pretty much independent, the ending will fall flat unless you get the reference to the first series, which will tear you up.
Some people like the huge Gundam series; I only know the short 6-ep series:
Gundam 0080 - War in the Pocket, which I love. I am told by people I trust that this encapsulates the very best of the whole franchise.
TTGL and
Diebuster have a lot of action, but
FLCL and
War in the Pocket are more domestic in scale, though still with robots.
Moving away from obviously shounen stuff, try
Puella Magi Madoka Magica (12 eps). Although at first glance a conventional magical girl series, it seriously subverts the genre, and pulls you in more and more. It also has some wonderfully fantastic imagery in the representation of the witches and the fights.
For a different kind of craziness, with a quiet, slightly snarky character (Kyon) as your guide, try The
Melancholy of Haruhi Suzumiya (two series of 13 eps), and then the film
The Disappearance of Haruhi Suzumiya. You can skip the second series if you want to avoid a series of highly controversial episodes which might put you off... I think they work - but you need the patience of a saint.
There is a time loop (known as "Endless Eight") which is represented by eight episodes, almost identical in content, but completely reanimated for each one. By the last you're ready to throw something at the wall - but that's the feeling they want you to have, and for me they judged it just right. I can't imagine having to watch them one a week for two months, as originally broadcast, though!
The film makes a very satisfactory conclusion (though the source novels continue, the anime doesn't, and will not), and is one of the best anime films out there - but it makes no sense unless you've seen at least the first series. Oh, and there are three orders for the episodes of the first series: "broadcast", "DVD", and "chronological". I watch in chronological order (combining both series), but there are good arguments for the other orders.
After all the fun and action and craziness, something really sweet and relaxing:
Haibane Renmai (aka
Charcoal Feather Federation) (12 eps). Quiet and sweet: you have to work out your own meaning for it. Perhaps it represents souls in the afterlife who need to accept the manner of their death before moving on to whatever comes next.
Finally, do you like something depressing as a change of mood? The most depressing anime there is:
Texhnolyze (22 eps). Very little dialogue (the first words aren't spoken until eleven minutes in, and we've already had a rape by then); violence, cruelty, despair, and in the end, nothing at all. Very powerful, but not for everyone.
Maybe that was too much all at once - but you can pick and choose (and there's plenty more where that came from).