"If you start the dams on Mt. Everest, you get more bang for your buck. And since you can make up any silly thing you want to support your experiment, you might as well go for it."

Rogun Dam:
"It is listed as the highest dam in the world - 335 metres high (1,099 feet)[2] - but this is a projected height. In reality the dam was only circa 60 metres high till 1993 and was destroyed by flood in the mid 90s. As of 2006 two projects are under consideration: the original, 335m, and an alternative one, 280-300 m, both having their advantages and drawbacks."
Nurek Dam:
"At 300 m (984 ft), it is the tallest dam in the world as of 2008."
http://en.wikipedia.org/wiki/Nurek_Dam

But I don't see anything about it's Hydraulic Height.

"A plate turbine has been developed for power plants under 4 MW. This type covers heads between 50 and 240 metres. The plate turbine is a simplified Francis turbine that has been developed in the university and research environment in Trondheim in Norway."
http://www.renewableenergy.no/sitepageview.aspx?sitePageID=1113

So with this kind of smaller turbine it can handle up a 240 meter head, but don't know the upper limits on bigger ones, but probably anything over 300 meters would be technologically challenging.