An application that springs to mind immediately is option evaluation.
Suppose I offer you to buy a contract to me: after three months I pay you the maximum value of the price of an asset.
How much are you willing to pay this contract ?
If you model the price of the assert by a drifted brownian motion, then you'll probably want to estimate the distribution of this maximum and take the expected value as a first guess of this maximum price. This will also be my first guess at the minimum price at which I will be willing to sell the contract. This is a non arbitrage price.
Note to purists : of course one will object that each of the parties can hedge himself, and that the distribution has to be corrected by a risk neutral argument (which will probably discard the initial drift and replace it by a zero risk rate drift instead).
N.b. : this is a real world application, exotic option traders buy and sell like contracts everyday.