Why does ?doing_wp_cron=***  show in my URLs.

There are a number of plugins and themes that append this to your URL. I believe it is typically used to perform a background process (like scheduled posts). WordPress redirects to the URL with ?doing_wp_cron  appended.

One such plugin, BackupBuddy, uses this to schedule tasks as a part of the backup procedure.

I don’t pretend to completely understand why it appends when visitors hit the site but it has something to do with the host or server having loopbacks disabled.  There are other plugins and themes as I understand that do the same thing.

Essentially it is trying to run a cron without actually running a cron.

In my situation the host has loopbacks disabled and are not willing to enable them. Many hosting providers will probably say the same thing unless you run your own un-managed VPS or have root access etc.

How can I remove ?doing_wp_cron from my WordPress URL?

It’s a sign that you have ALTERNATE_WP_CRON defined in your wp-config.php  Try to find and change the value of ALTERNATE_WP_CRON from TRUE to FALSE or add define('DISABLE_CRON', true);