I don't make a habit of it, but I occasionally turn the wheel too far and get an annoying noise - like last night when I got the Runner stuck in snow for the first time ever. I've read threads where some people have said that this is normal. Is it?

Mine seems to sing happily when Im at full lock to thr right....to the left though it more of a grunge melody
Ha!

No. At full lock, there's a very slight sound that would easily be drowned out by highway traffic noise.

I wouldn't know. Full lock is something I try not to do.

If its squealing odds are you are low on fluid, could only need a few tablespoons full to stop it.
Also means you probably have a slow leak, mostly likely on drivers side you will see a wet line coming from the steering pump.

Mine squealed at full lock. I'm assuming the squealing was the steering belt. Then I replaced all the drive belts when I did my Timing belt. It went away. So I'm guessing since the belt was old, it was slipping, or it was tensioned incorrectly.

This was the problem on my pickup. Check your fluid, the power steering cap has a hot/cold level on it. Then again 3rd Gen pickups make some awful noises at full steering lock, sounds like your front end just snapped in half.

Your steering will be louder at full rotation since your fluid is flowing hard and fast. You could try bleeding your steering per the FSM:

1. CHECK FLUID LEVEL (See page SR-5 )
2. JACK UP FRONT OF VEHICLE AND SUPPORT IT
WITH STANDS
3. TURN STEERING WHEEL
With the engine stopped, turn the wheel slowly from lock to lock
several times.
4. LOWER VEHICLE
5. START ENGINE
Run the engine at idle for a few minutes.
6. TURN STEERING WHEEL
(a) With the engine idling, turn the wheel to left or right full
lock position and keep it there for 2 - 3 seconds, then turn
the wheel to the opposite full lock position and keep it
there for 2 - 3 seconds.
(b) Repeat (a) several times.
7. STOP ENGINE
8. CHECK FOR FOAMING OR EMULSIFICATION
If the system has to be bled twice specifically because of foaming
or emulsification, check for fluid leaks in the system.

Thanks for the replies; that gives me a lot to work with. I probably won't investigate further until the weather is warmer because this problem rarely occurs, but I'm glad to hear that it's not supposed to make so much noise.