Exim is going crazy, how can I track down what it's doing?

If you have many exim processes, the first place to check is:
Admin Level -> Mail Queue Administration

Check for any patterns in the sender or recipient addresses.
Check the status of some of the message to see why they're in the queue.. if there is a common problem as to why they're in your queue (bottom textarea)

You can also get exim to create a stats page for you:

cd /var/log/exim
eximstats mainlog > stats.txt
less stats.txt

Check it to see where most of the emails are headed, either outbound or local.