My absolute favorites that I will rewatch and that while watching, I was thoroughly moved, regardless of whether it was a comedy or a dark story.
1) Nodame Cantabile - youth, enthusiasm, love for music, passion to be able express one's self in a creative way. If you like something clever & funny you will like it, but if you can't stand anime humor, then probably not because it follows the original manga very very closely.
2) Nobuta wo Produce - I'm not even kidding when I say I wanted to cry many times watching this drama although most of the time it felt like a comedy more than anything else. The most touching friendship story ever made! Besides, it's clever and also really funny! So many puns, subtle jokes and random goodness!
3) Byakuyakou - anyone who can't stand dark drama probably shouldn't watch this. Those who have never seen a dark drama should give it a try. Exposes the many dark sides of society, but much more than that, it's about a passionate and undying love between two people, whose love for each other really transcends that of a romantic couple.
4) Shiroi Kyoto - it's a medical drama, but far from being technical, it's very easy to understand. The medical terminology is minimal. Anyone can enjoy this drama. The plot revolves around two doctors, one who puts medical ethics above all, dedicated to his patients days and nights, and one obsessed with power and making it to the top of the University hospital. Somehow, they're able to respect each other despite not getting along very well at times. A touching drama about doctors - patients relationship and medical ethics.
Others that I enjoy
H2 (Yamada Takayuki, Ishihara Satomi)
Sekai no chuushin de, ai wo sakebu (drama version w/ Yamada Takayuki and Ayase Haruka)
Triangle (Eguchi Yosuke)
Proposal Daisakusen (not so much the actors, but I like the plot & motif)
Tatta hitotsu no koi (Kamenashi Kazuya, Ayase Haruka)
Liar Game
Engine
Long Vacation
Aikurushii