Maaaan, it's hard to narrow down only 5 shows, but here goes! 
1. I My Me! Strawberry Eggs - Fantastic show with an odd premise, and includes the best anime soundtrack I've ever heard in my life. <3
2. Kanon (2006) - Wonderful, wonderful, wonderful show. I recommend this to ANY anime fan who's not obsessed with blood, guts and violence. <3
3. Azumanga Daioh - The epitome of slice-of-life comedy. Watch it! 
4. Chrono Crusade - Great characters, awesome music, and my personal favorite "Action" anime. Rosette = <3
5. Ah! My Goddess TV - 50(+2) episodes of heartwarming harem-lite comedy/action/drama. What more could I ask for? :3 Belldandy to this day is still my Number One favorite anime girl!
... I was planning on putting some honorable mentions, but I would list at least 10-15, so I'll just stop here. 