Wouldn't the price of having a masternode keep going up then?
Yes, it will until it will reach some market-dictated price. In fact it is not a "price" because you don't loose your money, you just can't use them as long as you are running the masternode.
Mr. Spread, also would we be able to add some coins to our masternode after it has gone live to keep it in the game?
The way I'm implementing it right now, you will need to resend you funds (with some additional amount of course) to yourself. So you will need to shutdown your masternode and then start it again. This has the disadvantage that you will not receive payments for some time because there will be some period before your masternode will be elected again after shutdown.