I know there are lots of good combo DVD player and VHS tape recording
devices.  But what I'd really like is a unit that can RECORD on both
DVD and VHS tape, all in one unit.  Does such a thing exist?  What are
the options?  (Obviously I could get a DVD recorder separately from a
VHS recorder, but that's not the question.)

"6. Is there such a thing as a VHS VCR/DVD or Hard Disk/DVD recorder
combo?
As of this time (2003), there are currently no VHS/DVD recorder
combos. Panasonic has tentative plans to introduce one into the
Japanese market later this year. If well received, you may see a
similar unit on U.S. and European store shelves in 2004. "
http://hometheater.about.com/library/weekly/aadvdrecfaqb.htm

For what it's worth, Go Video's web site indicates that a "DVD
Recorder + VCR" is "Coming Soon."
http://www.govideo.com/