Here are some anime that I absolutely adore and rewatch with regularity. I wish that everyone gets a chance to see them! There are in no particular order.
Tonari no Totoro / My Neighbour Totoro (1988)
Just about anything by Miyazaki Hayao's Studio Ghibli is adorable. But My Neighbour Totoro takes the cake. A simple story presented with grace, skill and beauty and the voice acting is well beyond par. Mei's first encounter with Totoro is particularly enchanting.
Nodame Cantabile (2007)
A "rom-com" that revolves around classical music, Nodame is an endearing anime primarily due to the eccentric weirdo ingénue, Nodame, and her relationship with Chiaki, her straight-laced mentor. The support cast is hilarious and the music is memorable. In the same respect, it serves as a lovely introduction to classical/concert music.
The anime spans a couple of seasons along with additional specials. While the sequels are eminently watchable, the first season is simply fantastically made.
Hajime no Ippo (2000)
This is a coming of age boxing anime. While it's pretty long and rather formulaic, the main characters and their voice actors are amazing and the plot has a nice balance of action and comedy. However the clincher for me is the music which really gets you going.
The art might initially appear a little tacky. But it grows on you :)
This is simply hilarious especially if you are familiar with Japanese culture to an extent. There are lots of in-jokes and in many ways Lucky Star is a cute parody. Konata's stubborn indifference to the reactions of her "normal" friends who in many ways each share one of her vices is particularly endearing.
Hunter × Hunter
This is another well made "coming of age" anime in an imaginary world which revolves around a Qi-like equivalent named Nen. The main attraction for me is the society in which they live in and the general atmosphere projected. The characters are excellent and come with a degree of depth surprising in this genre. The plot-lines are intriguing and the battles - both physical and otherwise - are interesting. There are two versions of this anime and I've only seen the first one which came out in 1999.
Mimi o Sumaseba / Whisper of the Heart
Another charming Ghibli movie that is in many ways underrated. It explores adolescence, innocence and ambition in a way only Miyazaki can.
Akage no Anne / Anne of Green Gables
Another coming-of-age story based on Lucy Maud Montgomery's famous novels. I've seen the American TV version as well which was enjoyable. But this anime trumps it in many ways as it isn't limited in terms of time and IMHO, the (Japanese) voice actors used are far superior in quality. Cockles, heart, warmed.
This is an anime to be enjoyed for its atmosphere and art. It's simply stunning. The stories are fascinating and the settings are magical.
N.B. It should go without saying that all non-English works ought to be watched in their original languages.