Deflated tube: Is it me, or it is the rim tape? I just had flat tube #4.
Enough is enough.

Please examine the photo of my rim and the valve stem hole. Is the rim tape too far off of the edge? I had inflated the tire with more air, per the first post's advice responses. It had been working fine for nearly a week now. 

Suddenly, pssssss and it's a hole at the valve stem. AGAIN. Just like the last two times. So am I just not pumping it up enough -- but if so why did it last so long -- or is it the rim hole rubbing away at the valve entry area and eventually cutting into the rubber at the valve?

I don't mind pumping it up...I have to do it manually, and fear for overinflation and potential explosion...but I can certainly pump it to the proper volume, as I have been riding on this tube for several days now.
Rim strips, proper ones from a bike shop, last a long time and are immune to the wear shown on your picture. It looks like the strip left the hole exposed enough that the base of the stem was allowed to make contact with the rim. 

My advice is to take off the rim tape. Clean the interior of the rim with acetone. Take some fine grit sandpaper (1600-2000 grit) and gently rub it into the hole from inside of the the rim. That will help smooth out any possible burs. Then install the rim strip (not tape) and for one last good measure use a tube that is one size smaller than you usually do. It is curious how the tape shifted. Perhaps the stem wasn't perpendicular. At any rate, the smaller tube would decrease the chance of this happening with no ill effects.

Good luck!