When setting up a form completing event to send email to the admin when the form is completed, no email is received in my inbox. Did I forget to set something?

First I would check in the Form Completion Events drop down to make sure you have an event there.

Some of the common mistakes I have seen made are below:

1. Not selecting "Any Field Response" in the Dynamic Field question.
2. Not clicking "Update Event".  If you "Update Settings", it does not save the event.
3. Not sure that it matters, but make sure you have a valid E-Mail address in the from.

Can you send E-Mail from other parts of your site?  You can test this under Host Settings in the SMTP Server Settings.