It depends more on the size of the lot, but I wouldn't be surprised if it's at least 100 feet tall. I would imagine courthouses are pretty expensive since they have upgraded security systems and layouts.
But for comparison sake, the new courthouse in Austin which broke ground last month, will cost at least $102 million and have 8 floors. The published height before said it would be 135 feet tall. It'll have 252,420 square feet and occupy a full block.
You can read about it here:
http://forum.skyscraperpage.com/showthread.php?t=173556