I'll second Jody on the pre-turbo Mule vs. Thrasher, since I've also rode in both cars. Both handle very well (although I didn't get a "to the limits" cornering test, but both turned very confidently and predictably), although in pre-turbo form the edge in power definately went to the Thrasher. I've also rode in the Red Witch, and Charley pointed out that it took a more delicate right foot around the track as the greater low end torque compared to the Thrasher required a bit more care on corner exit technique.
The nice thing about underhood excess is that as long as you have a controllable powerband and good manners, the excess can be moderated without too much difficulty through either low tech means (right foot application) or high tech means (traction control, programmable boost controller, etc.) There is something to be said though about a well-developed combo that inspires confidence by not having you constantly on edge thinking "I hope this thing hooks..." every time you dip your foot into the throttle.
My car isn't on par with Jody's ZL1, but it isn't a slowpoke (expecting ~650 rwhp once I get the freshly broken rear end fixed and make it to a chassis dyno) and it does require attention to throttle input, especially in 1st and 2nd gear (both totally useless except for smokeshows) and occasionally in 3rd below about 65 mph depending on the road surface. So it's not a "carefree" drive at lower speeds and you have to use some caution in throttle application. However, it won't do anything stupid unless YOU stick your right foot in it. It is balanced though? I'd say yes. The powerband is nice and smooth and predictable and the engine's manners are excellent; it has no trouble at all putting around town with the A/C on, and so far I've seen a best of 24 mpg on the freeway going 80-85 most of the time. Just like Jody's ZL1 as long as the weather is OK and the car isn't apart for work I drive mine every day, either cruises out in the country or running errands in town. The car is perfectly happy doing either, although I'm much more happy out on open country roads.
Given what's on the market today, it is most definately possible to have very high levels of horsepower and still have the car somewhat "balanced" because with modern technology it's easily to dictate how and when that power is delivered which goes a LONG way to the overall balance and controllability of the car. As Jody and others have proven it is possible to have very high HP levels that 20 years ago were unheard of for a true street car, and nowadays that power can be delivered in very smooth, controlled ways. A turbo'd or supercharged (centrifugal, lysholm, etc) 900 hp engine of today with modern EFI is leaps and bounds ahead of say a 900hp carb'd and roots-blown engine of yesteryear in terms of driveability, fuel economy, and good street manners. You can have lots of power with none or very little of the "bad" behavior which have typically been associated with high-HP powerplants even 10 years ago. I find this especially true for forced induction engines, as when built right they will be VERY tame and managable when "off boost." You can go pretty darn crazy even naturally aspirated with today's technology and still have things very manageable and street friendly-- look at any of the 600+ hp big inch LS1's running around out there in Vettes and F-bodys and the like.
Tyler, I do have to agree that at a point too much power is detrimental to the overall balance of the car. You can certainly overpower your tires (or chassis!) but there are electronic aids to help compensate at least for the traction problem. In a way this is good-- the electronics will always deliver the maximum amount of power the tires can effectively utilize. If you have 900hp on tap and the tires can only take 550hp at a given time, the electroincs will do all the hard work for you. However, this could also slow you down if the traction control device suffered from severe latency issues, was too aggressive on the initial retarding, or too slow in ramping the power back in. So once again there's an argument for building the "right" amount of power into your car. I guess in the end you get a really impressive dyno sheet showing what it could do if it could completely hook and you will be on a first name basis with your local tire dealer. Although, that seems to keep many (I'd say most of us) very happy.
In the end, it's completely addictive and very entertaining to have the excess available even though you can never really use it all. Can it affect the balance of the car? Yes, but given the controlled manner in which power can be delivered with today's technology and also newer developments in suspension and tire tech increasing traction and handling limits I think the proverbial "loose nut behind the wheel" is more of a detriment to a car's balance than the powerplant, "overpowered" or not. It still needs to be a reasonably sorted combo to have potential, but it's been proven time and time again that a good driver behind the wheel of a mediocre car can turn faster times than a mediocre driver behind the wheel of a good car.
However, nobody is going to be winning any races behind the wheel of an 86 Yugo GV. :p
Troy