Most of these can be watched without knowledge of the culture, or rather, acclimation.
Fate/Zero
Code Geass
Overlord (the novels are great too)
Trigun
Aldnoah.Zero (only recommending for the tension. only up to episode 12.)
Hitsugi no Chaika (only saying this because of good plot)
Working!!
Steins;Gate
Shinsekai Yori (arguably shit)
Psycho-Pass
Read or Die
A little bit deeper:
Higurashi no Naku Koro ni
The World God Only Knows (read the manga for the whole story, its great)
Durarara!!
Hyouka
Renkin San-Kyuu Magical Pokaan (only watched because NEVER EVER, biased pick)
Amagi Brilliant Park
Full on weebshit:
Non non biyori
Yuru yuri
Strike Witches
Rokujouma no Shinryakusha (personal favorite, its really just haremshit but good)
I'm not sure where this falls on the rabbit hole scale:
Sayonara Zetsubou Sensei
Additionally, Neon Genesis Evangelion is a special type of shit. There are many reasons to enjoy/like/love/hate it, or not care about the characters at all, but it's worth a shot.
