The Love Live! Franchise - Aside from enjoying cute anime girls, I literally cannot come up with a valid reason based on the work alone that you should watch this. It's something you're either going to be swept away by how fun you find it, the music, the fandom, and the characters doing cute things over what can be a pretty cliche story in spots, or you're going to find it's not your cup of tea at all. That said, pretty much every episode has left me with a dumb grin on my face.
The Genshiken series - this is the story of a college "visual media" club from one student's enrollment to his graduation and a bit beyond. It has compelling characters, comedic punchlines and references, and is very well written IMO.
K-On! series - It's about an after school rock music club and follows our main characters through their 3 years of high school. It's got some pretty good music and some charming moments, as well as quite a bit of layered storytelling and actual growth even if it feels like a very slow-paced show compared to what many people here seem to enjoy. Most episodes leave me with a warm, fuzzy feeling with a dumb happy grin on my face. I think the only time I haven't was when I completed the movie as that meant there was no more animation left of these 5 girls and their teacher. I felt a void upon completing this, that's how much of an impression it left on me.
Non Non Biyori series - If you thought K-On! was too slow, this one is probably not for you, either. This is arguably even slower paced as its two seasons center around different standard life events that happen over the same year - the second season is literally titled "Repeat" to indicate it happens concurrently and even interweaved within the storyline of the first season. This is a show that exists for healing and is part of the Iyashikei (healing) genre. It's a slow paced show that's calming and has a charming cast. Pretty much every episode has left me with a warm, fuzzy feeling and a dumb grin on my face. I think only the last episode of Repeat did not but that's because it was the last of the adaptations at the time I completed it.
R.O.D. The OVA - There were a few series I could have picked, but for me, this was the easiest to write about. It's a show largely set in the modern day, except that there are a bunch of spies working for a secret agency that have supernatural powers, and there are some battles involving other spies and a few surprises that are totally worth it. This 3 episode OVA is an adaptation of one of the volumes of the novel series by the same name. It will require some suspension of disbelief but it should be a very fun ride.
And that's way more writing then I've done in quite a while. Took nearly half an hour to write all this up.