>>877449FLCL is amazing (top 10 favorites for me), S2 is ok but nothing S1. S3 is trash.
Samurai Champloo is probably my favorite of all time, its episodic so you be binge watching it but its really good. This and FLCL have the best OST ever.
Welcome to NHK is good if you like some drama about a depressed NEET. Good OST
Code Geass for some mindless fun
Death Note for great suspense and OST
Monster for a long mistery/drama that really pays off in the end. Best ''villain'' ever.
On top of mind those are a few I rec. There are a lot of early 2000's anime that are completly 'unkown' or not discussed. 2006-2009 is probably the golden years of anime.