How can I re-arrange sharing application icons? When I click on Share from within an application, say an RSS reader, I will see a multitude of choices. See screenshot. 



How can I re-arrange it so that my often used applications such as Hangouts and GMail can be moved to the top of the list of applications?
With the built-in "Share" menu, that's not possible AFAIR. However, there are alternatives available – I know of two: AppChooser and AndMade Share:

  
AppChooser (list view / grid view) and AndMade Share (source: Google Play; click images for larger variants)

I'm using the paid variant of AppChooser, and am quite happy with it. As above screenshots show, you can sort the list alphabetically. You can also hide items you don't need (they still show up if you select "ALL" at the bottom). If you didn't order them alphabetically, items are listed in a "last recently used" manner – which keeps your frequently used apps at the top of the list automatically. Works great, I'm using this for quite a while now.

AndMade Share has a paid variant as well. In the third screenshot above you can see its advantage: you can sort your list manually as you like. Which one to chose, is rather a matter of taste :) Both hook into the "share intent" – so on your first "share" after installing one of them, you will be asked whether to use the built-in share menu or your new alternative. I did so, and made AppChooser Pro the "always use" default variant – never regretted that or turned it back :)