How many times can really I share a purchased app with other people
The limits are only enforced on the authorization level. The license terms of the app store are very confusing to me (I have to read them many times and I never know if I'm remembering some past wording or what it really says today). It also has all the various stores - each with different terms for rentals, movies, music, subscriptions, Mac Apps, iOS apps, business use and personal use.
http://www.apple.com/legal/itunes/us/terms.html
One computer that is authorized to download apps can sync over USB to hundreds of devices. I have never heard of a hard limit to the max number of devices per computer. Via USB, you can share any or all your apps with countless people. This is called out in the current license terms as follows:
(i) You may download and sync an App Store Product for personal, noncommercial use on any iOS Device you own or control.
If you sync in manual mode, you have to designate this account as the "primary" account. That's it
There are further restrictions when you want transfer that app to another computer or update the app. You don't have to register the iOS device containing your app unless you want to update over the air ( OTA ) from the app store instead of from the computer you control.
Each device is said to have a 5 account limit ( according to the license wording - i've never tested it to see this happen in practice)
Your Apple ID likely has a 10 device limit - 5 of which can be computers. This limits your ability to share your Apple ID. ( but not the apps themselves )
I haven't poured over the store license so keep in mind that you might have more technical capability to share than your license permits. I am also not a lawyer - so I don't know if two people can be considered to "own or control" the same device. Just like you can drive without a license - you can operate software without a license. Be nice and know what you are getting in to.
It's based on the honor system. There aren't any actual [software] restrictions on the number of devices, but legally speaking, you are supposed to refrain from installing on any devices that aren't your own. That 5 person limit is strictly for iTunes [the desktop app].