How can I see the phone number dialed after I initiate a call? In the Phone app (iPhone 5, iOS 6.1.2), I click on a contact's number, the call is initiated and starts ringing. 

For a brief moment at this point, the label of the number (home / work / etc…) is displayed alongside the contact's name, but it quickly is removed and replaced with the call timer counting up and only the name (and picture of the contact) are shown.

  

How can I see the number that is being called during a phone call?
You can't - in a straight-forward way. 

The best work-around is to tap the "Contacts" button on the call screen. This will bring up your list of contacts which you can then select to see the number(s).

The good news: iPhone remembers you position in the contacts when you initiated the call so you don't have to scroll down all the way from "A" contacts.

The bad news: Hopefully you don't have more then one number for a contact. If you do (and most probably do), there is no way to know which of those 2 or 3 numbers you choose when you called the contact.

Interestingly enough, iPhone is nice enough to tell you which (of several under a single contact you are calling) number you are calling when you add a second call. Apparently as of iOS 6.1.3, iPhone no longer does this.