XenForo is without any doubt my favourite piece of web software available, I have several licenses for it and have made good use of every one of them and would consider myself pretty good with it, it’s not often that I don’t work out a problem with it now – though I have been using it for quite a few years.

I have found a few addons that have stuck by me and never failed me - do note that these are all for XenForo 1. I thoroughly recommend using XenForo 2.x now that all of these are implemented in some way. If you choose to use XenForo 2.x, here's a list of my favourite addons for that.

All Rich Usernames
I find that by default XenForo has usernames quite dull in most areas, this addon is something that should have been implemented already and is such a small implementation however I find it crucial.
https://xenforo.com/community/resources/all-rich-usernames.2994/

Last Post Avatar
Last post avatar is one of those addons that is starting to be made the norm, with companies such as Audentio implementing it in to their styles and I believe it’s planned to be added to XenForo, it adds a small avatar to the left of the most recent post in a sub-forum on the forum homepage and just makes the forums less dull to look at initially.
https://xenforo.com/community/resources/th-last-post-avatar.3921/

TaigaChat
Most forums have a shoutbox, I’d probably only recommend this if you’re bringing in enough traffic to warrant losing some threads to a Shoutbox. There’s a new Shoutbox out that a lot of people have been switching to, however I’d rather stick to the one I know works.
https://xenforo.com/community/resources/taigachat-pro-realtime-chat-shoutbox.1224/​

MassBanEmails
This is something I have to install to prevent basic spammers using throwaway email accounts, you’ll need to find your own ban list however as mine is private. This list may do you well.
https://xenforo.com/community/resources/mass-e-mail-address-banning.4267/

[bd] Widget Framework
Though there is essentially no support from the author for this addon, it’s free and I’ve never had it break so I can’t complain. XenForo need to implement a widget framework in to xF soon, because it’s a crucial part of all of my forums to make them stand out and add widgets everywhere.
https://xenforo.com/community/resources/bd-widget-framework.297/

Who Has Visited In Last 24 Hours
I believe this has been deleted, so you may need to look elsewhere for a copy but this is a great little addon to show whos visiting throughout the day.
https://xenforo.com/community/threads/who-has-visited-in-last-24-hours-paid-deleted.44838/​

PE Forum Statistics
This adds vBulletin-like forum statistics and members online to your forum and removes the default widgets on the sidebar. I personally like it, however it’s not for everybody.

I have since re-made this addon using only templates as I found an incompatibility with one of my addons - but it's still worth getting if you're not able to do that.
https://xenforo.com/community/resources/forum-statistics-pixelexit-com.2577/