Yuki Hase has notice that a girl in his class, Kaori Fujimiya, spends a lot of time alone and has no friends, so he decides to try to become friends with her, and with a little persistence after an initial refusal, a bit of a friendship develops between them. But as the school week ends, Kaori tells him the reason why she doesn't really want to be friends with anyone. She suffers from a strange form of amnesia. For a past few years, every Monday morning when she wakes up she has lost all memories from the week before of her friends and any time she has spent with them. Hase isn't sure if this is true, until he meets her at school on Monday and sees that she really has no idea of who he is, and he has to start over again in winning her friendship.
A few weeks go by, and with the help of a couple of other friends, Kaori begins to open up around others in the class, making new friends, and her memory seem to be improving. Then someone who knew her when they were children shows up again, and seems to be angry with her for some strange reason. Meeting him causes her memory problems to relapse, and she again forgets who Hase and her other friends are. Fighting discouragement, Hase continues to try to help Kaori, and he eventually finally learns what really happened on the day that Kaori's memory problems began.
Friends of Friends
A special mention should be made of the two friends of the main characters, Shogo Kiryu and Sagi Yamagishi, whom I respectively and affectionately think of as The Vulcan and The Space Cadet. Shogo is a friend of Hase's when the series begin, and his serious and logical demeanor provides some support for Hase's more up-and-down temperament. Sagi is a girl who shows up a few episodes in, and her own more conventional forgetfulness, laid-back personality, and occasional ignoring of other's personal space does give the anime some light-hearted moments.
I don't want to go too far in comparing Hase's concern for Kaori to what the Bible says about love in I Corinthians 13, but a bit of a look might be good. And just to be clear, I'm not using “love” here in the strictly romantic sense, but more for a concern for another person's good.
For example, let's take “love is patient and kind”. There are times when Hase displays these things very well. He tries to be very understand of Kaori's difficulties, both with her memories and with the problems that can cause other people. He is the one who suggests that she start keeping a diary that she can read to remind herself of what she's forgotten, and even spend considerable time and effort trying to find it for her when she has lost it and forgotten about it.
On the other hand, he has bouts of jealousy, too. He wants Kaori to be like a normal student with lots of other friends, but as she begins to make these new friends, he begins to fear that his friendship with her is becoming less special. This jealousy causes a bit of conflict early on between them.
Or look at where I Corinthians 13 says that love bears, believes, hopes, and endures all things, and then look at the last couple of episodes, where they learn what happened to Kaori as a child that caused her memory to be bonkers. At this point, Hase stumbles a little, as he gives in to a fear that somehow him being close to Kaori might hurt her and lead to a return of her memory problems. This does get corrected, and the series ends on a nice, satisfying note.
Anime doesn't have the best of reputations when it comes to hot it handles these kind of stories, and to be fair, the problems are often too real. Gratuitous fan service, hyper-sexualizing characters who are little more then children, annoying harems, and other tropes are far too common and far too distasteful.
So, it's good to say that such things are absent from this series. Instead of relying on those things, this series has a solid story with believable and sympathetic characters. True, Kaori's amnesia does seem contrived, though I know mental problems can be very tricky so I can't say this kind of thing has never happened, but if you can suspend you disbelief concerning her memory problems, then I think you'll have few other problems.
I give this series a very strong recommendation.