The exact amount is never stated in canon, and it's impossible to estimate without knowing exactly how many Galleons, Sickles, and Knuts Harry owned. But it was a considerable amount:
Griphook unlocked the door. A lot of green smoke came billowing out, and as it cleared, Harry gasped. Inside were mounds of gold coins. Columns of silver. Heaps of little bronze Knuts.
"All yours," smiled Hagrid.
All Harry's -- it was incredible. The Dursleys couldn't have known about this or they'd have had it from him faster than blinking. How often had they complained how much Harry cost them to keep? And all the time there had been a small fortune belonging to him, buried deep under London.
Hagrid helped Harry pile some of it into a bag. "The gold ones are Galleons," he explained. "Seventeen silver Sickles to a Galleon and twenty-nine Knuts to a Sickle, it's easy enough. Right, that should be enough fer a couple o' terms, we'll keep the rest safe for yeh."
Harry Potter and the Sorcerer's Stone - pages 58-59 - US Hardcover Edition
It is estimated Harry had quite a lot of money, which he inherited from James and Lily. I'll see if I can find an estimate of Harry's wealth in Muggle money. Was he extremely rich? Yeah, I'd say he was quite wealthy, but probably not as wealthy as the Malfoys. There is no canon evidence that there were any charitable funds set up on Harry's behalf to receive funds for him. I'd say that would be pure conjecture.
ETA: The Harry Potter Lexicon has the following:
JKR has stated in an interview (Comic Relief 2001) that she estimates the value of one Galleon to be "about five pounds," which works out to around US$9.75 (the exchange rate at the time of the interview was US$7.33). In the introduction to both QA and FB, US$250-million is stated to be the equivelent of 34 million Galleons. That also works out to a value of £5 to the Galleon, at the exchange rate of the time. The price listed on the back of the books, however, is not correct, since US$3.99 would equal less than half a British pound, or 8 sickles and 15 knuts. The book instead incorrectly lists US$3.99 as being equal to 14 sickles and 3 knuts. (Unfortunately, CNN.com uses this incorrect value for their Knuts-to-dollars converter.)
This gives us approximate values as follows:
1 bronze Knut = £0.01 (US about 2 cents)
1 silver Sickle = £0.29 (about US$0.57)
1 Galleon = £5.00 (more or less US$9.75)
These rates vary as the exchange rate fluctuates - see the Wizarding World Currency Converter for the current rate.
There is apparently some kind of foreign wizard money that consists of gold coins the size of hubcaps (seems impractical to me, LOL!). So if we had any way of knowing exactly how many Galleons, Sickles, and Knuts Harry had, we could estimate his net worth. Unfortunately, I can't find a specific number quoted by a reputable source, but admittedly I didn't look super hard. :)