How many bets do you put in on the river?

[PygmyHero]

I was playing on the Cryptologic network this evening, and if you're not aware, the site allows unlimited raises and re-raises on the river if the players are HU, until one or both are all in (this rule may also apply on other streets, I'm not really sure).

Two hands arose where I found myself HU at the river with a strong hand, wondering how many bets to put in (let us assume for the purposes of this exercise that your opponent will NEVER stop re-raising, meaning you can get all in or just call at any point).

Hand 1: I have KQo and raise first in from the CO. Only the BB calls. The flop is 3KQ rainbow. BB bets, I raise, BB calls. The turn is a 6. BB bets, I raise, he calls. The river is a Q. BB bets, I raise, BB re-raises, I four bet, BB 5 bets, etc.
After the 5 bet the BB has 30 more big bets, and I have him covered. Note that I have the second nuts and only lose to KK. No read.

Hand 2: EP limps, I raise in MP with AKo. The SB three bets, EP calls, and I just call. The flop is KQ4. SB bets, EP calls, I raise, SB 3 bets, EP folds, I call. The turn is an 8. SB bets, I call. The river is a K. SB bets, I raise, SB 3 bets, here we go...
After his 3 bet the SB has 23 more big bets, and I again have him covered. I have been with this player for a while, and my note on him says that he is very aggressive, and pushes very hard with almost any playable hand. I have seen him caught with TPNK on several occasions. My read PF was AA, KK, or QQ, with AK being a smaller possibility. A weaker hand is possible as he is a little maniacal, but he IS in the SB. Post flop I felt AA or QQ was the most likely (note for KK he would have to have the case kings). On the river it is obvious he cannot have KK and I am now beating AA, but still lose to QQ. I also lose to a K that hit it's kicker (KQ, K8, and K4) and split with AK. I think the chances of AK or a K that has hit it's kicker is very remote. Note that my hand here is fairly strong, but not anywhere near the nuts.

I'm not looking for a critique of any of my earlier play, I simply want to know how many bets you would put in on the river (again, assume your opponent will re-raise until he is all in). I know the answer is 'it depends' on what you're comfortable with, etc. I would still like to hear some people's thoughts on this, and their reasoning.

For the curious, results are in white below:
<font color="white">
Hand 1: Villain has 33, good for threes full of queens.
Hero has queens full of kings and takes the pot.

Hand 2: Villain has K9o, good for trip kings with Q9 kicker.
Hero has trip kings with AQ kicker and takes the pot.
And yes, I did have the note on him before this hand.

w00t! </font>