The idea that you can only fold paper eight times in one direction rings absurdly false, since the problem screams of being obviously related to the thickness of the paper in relation to length.

An informal, physical/graphical proof that there is no upper limit given sufficient length is simply this. Imagine a piece of paper that has already been folded N times:

           --------\
          |         \
          |______    |
          /------`   |
          |          |
          \---------/
             
         ->|      |<- flat section

All you have to do is add more material to the flat section, extending the length of each layer by an equal amount, and then you can fold it again:

           ------------------------\
          |                         \
          |_____________________     |
          /---------------------`    |
          |                          |
          \-------------------------/
             
         ->|                      |<- flat section

You can add as much as you want. Suppose the above is an inch thick and we extend it to be a mile long.

I.e. don't approach this as a problem of the subdivision of a fixed length and width, but rather as the length being open-ended: we can go back and add more.