apt-get produces the error; exim: failed to find uid for user name "Debian-exim". Exim is an MTA or mail transport agent. Do not suggest I use Postfix please.

[Nate Homier]

The meta package exim4 creates the user Debian-exim, so exim should
know that user.  When removing exim, I think the user Debian-exim
get's left behind, meaning it's not removed when removing exim4
package.