"Fill up" usually denotes occupying space and the "up" is optional. For example, you can say "I don't want to fill up your brain with ideas" or "I don't want to fill your brain with ideas". Also, "I need something to fill up my brain with ideas" or "I need something to fill my brain with ideas". Both work. *However, one thing to note that other people might note is that when you say "fill up", it emphasizes how full something is, whereas if you just say fill, it is usually implied that something is fully filled, but not always.

The "up" in "fill up" suggests a completed action. Your brain becomes full and there is no space for anything else. In your example, either "fill" or "fill up" could work.

Saul is right, but if you did want to use this exact sentence, I would say "I need something to fill my brain". But what is it you're trying to say exactly? Are you looking for a distraction or are you trying to fill your mind with knowledge? If distraction is the case, I would say "I need something to occupy my mind" :)

© "I would complete an application form. I might fill in the application form. I would prefer NOT to fill out the application form. Despite the silence, BE does not really like filling out. Most, but not all of us, fill in or complete, forms. " ©