My definition of motivational anime is that it has character who has strong mentality, works really hard, overcomes his weaknesses, tries to achieve a big goal, and ultimately is rewarded in the end.
I think this type of anime is especially useful when you are trying to learn something or needing some encouragement to achieve a certain goal (e.g. passing exam), or needing some adrenaline boost, or even when you are bored with those harem, romantic or comedy animes.
Most of the motivational anime is of shounen adventure and sports genre, where the main character has dream that they want to reach. So, here is my list of motivational animes :
Wants to get stronger to be able to eat more and more delicious food. Why not? There are lots of feasts (rewards) waiting.
Baseball anime. There are 5 seasons, of which all of them are motivational. I recommend watching the 3rd season when Goro is leading a no-good baseball team to win the championship.
A kid from poor family trying to win racing championship, which costs a lot of money of course.
4. Shijou Saikyou no Deshi Kenichi
A bullied guy in the school, due to (un)fortunate circumstances, undergoes training by the strongest teachers around.
5. Hajimo no Ippo
Boxing anime. Training hard, trying to win a boxing fight with no regards of all those injuries. Most of the (good) boxers are very motivational. I recommend watching Takamura vs Hawk episodes.
Two young students having a dream to become famous mangakas. Lots of trials await along their way.
7. Space Brothers
Siblings who are dreaming on going to space since they were little. This is a light hearted anime and full of comedy too. You can feel a lot of hopes and dreams in this series.
8. Sakurasou no Pet na Kanojou
This anime in a glance is your kind of usual romance/comedy anime. However, the message and plot that drives this anime is very motivational. Most of the main characters are working hard towards their goal while trying cope with daily lives.