My guess was:
- Plugins
- WordPress core
- BuddyPress
- Themes
but, I’m curious what you do on any sites you administer. Keeping WP updated can sometimes be a chore — especially if you run a network of sites or use many plugins, BuddyPress, or themes with a lot of functionality!
Assuming you are backing up before updating anything I would go with:
and make sure to test thoroughly after each step.
The order above is going from that which has the potential to screw up the most things (core) down to the least (themes, usually). Because Core is the step most likely to break all other things I’d rather get it out of the way and know if this is going to be a “cup of coffee day” or an “order in and bring a sleeping bag” day. If my sites survive a Core and BuddyPress upgrade back-to-back it’s unlikely that a mere plugin or theme update will bring me down completely (famous last words, right?).
In reality, there probably isn’t a single, correct answer to this as it depends on the number and types of plugins and themes. If your stuff is well-coded or coded by people who know what they are doing then updates should not be traumatic events. On the other hand, if you are just slapping the first thing that catches your eye into your site and not checking the code and history, then you should be prepared for problems and be able to roll back.