Can a Mac Mini have 3 external monitors connected to it?
Yes, a mac mini can have multiple monitors connected to it. There are now very good USB to Video solutions on the market that will correct this limitation.

Steps:


Choose a good Mac compatible USB to DVI or USB to VGA adapter. There
are many available from different manufacturers. Look for adapters
that contain the DisplayLink chip as these are Intel-Mac compatible.
Make sure you've downloaded the latest DisplayLink Mac Drivers, like
these: http://sewelldirect.com/Drivers/DisplayLink-OSx-1.1.zip
Install the drivers through the guided setup by first selecting the
image file (.dmg) and opening it, accept the license agreement,
select which drive to install the driver on (Macintosh HD), press the
install button, and restart your computer.
Plug your USB to Video adapter into your computer's USB port and
attach your monitor's cable to the adapter. You can do this while
your computer is on and it will recognize the device in the same way
it recognizes most plug-n-play devices.
Once your computer recognizes the adapter, it should automatically
extend your desktop. If you prefer that it mirror the original
monitor instead, or if you want to change any of the resolution
settings, go to the Systems Preferences area and select "Displays".


When setting up multiple monitors choosing two of the same size is recommended.

Source.