How many device can install a purchased apps?

Welcome to the Support Communities.

You can authorize up to five computers to use your purchased items. (Source)

But see also this thread and this thread.

Additionally, if you are talking about apps for computers purchased at the app store, you can install them on as many Macs as you own or control. Excerpt from the software license agreement of iMovie (as an example):

Permitted License Uses and Restrictions.

A. License.

Subject to the terms and conditions of this License, including but not limited to the

requirements and limitations in Section 2B, unless you obtained the Apple Software as described in

Section 2C, you have the right to use the Apple Software (whether you obtained it from the Mac App

Store or preinstalled by Apple on Apple-branded hardware) as permitted by the Mac App Store Product

Usage Rules set forth in the App Store Terms and Conditions (http://www.apple.com/legal/itunes/ww/)

(“Usage Rules”), and are granted a limited, non-transferable, non-exclusive license

to download, install, use and run for personal, non-commercial use, one (1) copy of the Apple

Software directly on each Apple-branded computer running OS X (“Mac Computer”) that you

own or control;