I haven't seen a recommendation yet for "
Bake Neko" or its sequel
Mononoke, so I'll toss them into the mix, along with very strong seconds for
Moribito and
Dennou Coil (yes, it has kids, but you should watch it anyway).
Some other shows that also feature adults:
Nodame Cantabile
Bartender
Hataraki Man
Usagi Drop
House of Five Leaves
Hyouge Mono
Mouryou no Hako
Mine Fujiko
Yakushiji Ryoko no Kaiki Jikenbo