I don't even know the name of planet I live on
I don't even know the name of the planet which I live on
I don't even know the name of the planet, I live in
(This was said with a comical sense)
I don't even know the name of planet I live on
I don't even know the name of the planet which I live on
I don't even know the name of the planet, I live in
(This was said with a comical sense)
Since you're talking about a specific planet, you need to precede it with the definite article the. Similarly, you live ON the planet, not IN the planet. If you say that you live in the planet, that would mean that you literally live inside a planet. Would that make any realistic sense? So far, the following example would be correct:
I don't even know the name of the planet I live on.
The use of the pronoun which is up to you in this case because it's oftentimes optional. You can even substitute it with the conjunction that. Their primary function is to link a clause giving further information to something that was previously mentioned:
I don't even know the name of the planet which I live on.
I don't even know the name of the planet that I live on.