How do I show only 3 of the latest posts for each term in a view?
If you need only one node you can use “Representative node” relationship. But if you need three nodes per each term, use something like Views Field View.

1) Create "child" view for content. Something like this:


Add taxonomy term argument, pager limits, sorting and etc:


Now you can attach this view to another one.

2)  Create taxonomy view:


Add hidden term ID field and then add “Global: View” field:


Don't forget to configure caching. From Views Field View page:


  It's highly recommended to use this module in conjunction with views caching.
  Views Content cache and Cache Actions are good ways of caching views.