There's loads of really good anime out there without a single robot or tentacle. Here are some personal recommendations:
-Azumanga Daioh, as we've already mentioned. It's licensed by ADV and they've realeased the whole series on 6 DVDs. There's a special box set out with all 6.
-Mahoraba ~ Heartful Days ~. I simply can't recommend this one enough. And the manga's even better. It's not licensed, so you can get it
here. I recommend Froth-Bite's sub.
-Air (
fansubs. I recommend either Lunar or the Triad's). Just buy lots of tissues beforehand.
-Kanon (
fansubs. Get Ishin's). Also stock up on tissues.
-Honey and Clover (
fansubs. Get Solar's). A really wonderful series with some beautiful artwork and a really strange OP animation. The 2nd season has just started (
here).
-Bottle Fairy. It's funnycute. Licensed and out on 2 DVDs.
-Kare Kano. Also out on DVD. The only thing is the company that was making it ran out of money near the end of the series and the season was never really finished. You can tell when all the flashback scenes start taking up like half the episode.
-Pita-Ten (
fansubs. Get Hikari no Kiseki). More funnycute but with more substance than Bottle Fairy.
-Aishiteru ze Babay (
fansubs. Lunar). This series actually made me want to have children at some point, that's how good it is.
-Ichigo Mashimaro. Released in the US as "Strawberry Marshmellow." The first DVD just came out not long ago. It's kinda like the elementary school version of Azumanga Daioh.
Well, that's a pretty good list, I think. If you want to see all my recommendations, go here:
http://www.japanesepod101.com/forum/viewtopic.php?t=405