I've been a fan of anime ever since I first saw Battle of the Planets (G-Force/Gatchaman) as a kid. Course I didn't recognize it as anime then and it wasn't until I saw Robotech as a teenager that I realized there was a whole genre of the stuff.
I haven't really watched anything new over the last few years but I still have my personal favourites. To this day I'm still a huge Gatchaman fan (own all of the original episodes of the 1st series - untranslated without subtitles).
Other stuff I enjoy is (not in any order):
§ Lupin III tv series & the movies (especially the Castle of Cagliostro movie)
§ Anything done by Hayao Miyazaki (Princess Mononoke, My Neighbour Totoro, Laputa: Castle in the Sky, etc.)
§ Bubblegum Crisis (Bubblegum Crash wasn't as good but is okay)
§ Ranma 1/2 tv series and some of the OAVs
§ Akira (but can't watch it any more -- seen it too many times ;p )
§ Golgo 13: The Professional (like the manga more though)
§ Wings of Honneamise
§ Phantom Quest Corps (Yuugen Kaisha)
§ Vampire Princess Miyu
§ El Hazard (fantastic series)
§ Gunsmith Cats (not great, but fun)
I've been watching anime now for over twenty plus years so I've seen a lot of stuff that I have now forgotten about and some stuff I wish I could forget about.
Cheers,
Tim