On http://www.BrainDash.com/community/, we use a number of plugins.  Some we've written ourselves, some we've obtained from the community.  Our own plugins, CoReg and AutoLogin provide us with a single sign on.  friend_request makes friends a 2 way thing, rather than the default (which is more like 'fans' than friends) and sticky_widgets means I can, as an administrator pretty much control what profiles look like.  That's hugely important as I may well disable some widget plugins and replace them with different ones and I don't want users to see the dreaded broken plugin widget. 

I've also written a few proprietory plugins which aren't available to the community, BD_Terms which changes certain core terms in the languages pages, that has to be the first plugin.  BD_Groups, which provides XML-RPC and XML interfaces to groups, BD_Scores which adds a widget to profiles to show your scores on our Third Party Site and BD_theme, which changes the stylesheet and a lot of the elgg view's to look like its part of www.braindash.com.

I'll let you know when I have a week spare. -.-

adultslike.me uses so many plugins and I don't really think we could do without any of them. I've modified most of them in some ways, though, so the functionality (either backend or frontend) is different to most other sites...

I have a folder crammed full of diff files so I can keep pace when there are updates to be had.

It's almost a nightmare. Except I enjoy it. :|

However, back to the topic...

vazco's topbar plugin is probably the most useful. By far. It's allowed me to use elgg's builtin features - groups, pages - to a degree I wouldn't otherwise. 

I also love izap_videos, tidypics, and autodash. :)

I'll come back and add links when I have more time, if needed.