I'm wondering whether there is a way to see all iPhone App Store purchases in a similar fashion that is available on Mac App Store in Purchased section.
This is now possible since iOS 4.3.3 by going into the iPhone App Store > Updates > Purchased. It provides a list of your iPhone app purchases and allows redownloading from the iCloud.
Also you can visit reportaproblem.apple.com and all recent purchases will be shown.
I don't think there is a way to view just iPhone App purchases. You can view your whole iTunes purchase history. You can see your iPhone app purchases there.
Open iTunes -> Store menu -> View My Account -> Sign in -> Go to Purchase History
Update: This seem to include all your purchases using your Apple ID (iOS App Store, Mac App Store, and iTunes Store)
On iOS, go to Settings > [your name] > iTunes & App Store.
Not true... None of these links are possible, nor if they when they do seem to be relevant show history of purchases. I do have doubts that Apple Store have charged me few times for an application I have paid for over and over again. So when it comes to check what have purchased there is no way possible, except on the phone and they are not in logical order, which makes hard to pinpoint which have I bought recently that I have gotten charged for it few times. Sometimes it shows that I have purchased something but the some don't show I have paid for them, although I did already. Surprisingly enough sometimes an application will show a price on it, I will install and it will say that I paid for it before, but other times more often it doesn't say that, and it makes me pay for it few times. Now I have been trying to look up the history purchase and no where in hell is it clearly apparent that I can achieve that mission.