lease up, how much can they raise my rent

Yes.

You have a lease for a certain period of time. Then you have no ongoing obligation to the landlord or the landlord to you. The landlord can kick you out and not offer you a new lease. You can choose to move out even if the landlord reduces the rent to zero. No obligations. But they could be interested, you could be interested, so there's a negotiation on price. They can't raise your rent because you don't have to stay - but they can propose a rent and you can accept it or reject it, or you can propose something lower and they can accept it or reject it.

>Can they legally raise the rent to anything they want?<

Yes.

yes