What is the worst an attacker could do with full control over the blockchain? Let's assume somebody gains control over 100% of the hashing power and is now free to mine and control as many new blocks as he wants. What would happen?

The attacker still could not include any fake transactions into his blocks, because other nodes could easily detect that the transaction wasn't signed by the wallet owner. 

The attacker could include some double spends from his own wallets, but again, this could easily be detected.

So, the other nodes would simply ignore his blocks as invalid. Maybe they would crash and require a fork, or they would recover and wait for the next valid block to be mined and broadcast.

It seems to me that there isn't really anything to gain from controlling the hashing power. Am I msising something?
You're correct that a miner with 100% control could not forge transactions, the nodes would just ignore them. They could perform a '51% attack' though, trading coins for another asset (confirmed in 'block x'), and then going back to re-mine a new 'block x' that does not include their transaction (so now they control the asset they traded for, and have their coins back. The network could catch on to this, but nothing 'invalid' will have happened). 

A miner can censor transactions though, so they could just mine empty blocks endlessly, effectively stalling out the network. 

Perhaps more profitably, they could set a very high minimum fee for a tx to be mined, essentially using their monopoly to extort users for higher fees. This could be done covertly so as not to raise suspicion: the miner could generate their own high-fee transactions to fill up a portion of the block, forcing other users to pay higher, or be excluded. The miner would receive 100% of their fees back since they control 100% of the hash power, but it would be tough to tell this is happening from the outside.