Princess Alexandros XVII said:
Hard to answer. Like many things, anime has various genres. What type of TV shows does the person like? That way you can recommend something similar.
This.
Also, what genre does this person enjoy in terms of various media: comedy, tragedy, romance, action-adventure, gore/horror? I can only talk about the anime I loved because they were very much well done (and in no particular order).
1)
Fullmetal Alchemist/Brotherhood - A shonen fantasy story about how magic or alchemy developed instead of science.
2)
Monster - A psychological thriller about a doctor who must figure out who is behind the deaths of several people before the killing continues.
3)
Ghost in the Shell/Stand Alone Complex - Scifi futuristic story about a secret team of "police" dealing with two specific investigations (in both seasons, the movies vary in plot).
4)
Samurai Champloo - A historical retelling of a story set in edo-era Japan about a young girl looking for her father. She comes across two ronin, and convinces them to help her on her journey.
5)
Welcome to the NHK - Story of several young adults living in Tokyo who are stereotypically categorized as hikkikomori or otaku. Lots of paranoid delusional stuff coming from the main character. Very funny.
6)
Cowboy Bebop - Another scifi story set in the future about a team of bounty hunters or "cowboys". Lots of character-driven storylines, very powerful and very well written and performed.