Exponentiation Sequence
05AB1E, 12 11 10 bytes

Supports the sequence up to 2^95 = 39614081257132168796771975168

₃ÝoŒʒg≠}Jù


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Explanation

₃Ý            # push range [0 ... 95]
  o           # raise 2 to the power of each
   Œ          # get a list of all sublists
    ʒ         # filter, keep elements that satisfy:
     g        # length
      ≠       # false (not equal to 1)
       }      # end filter
        J     # join each
         ù    # keep numbers of length matching the input


Saved 1 byte thanks to Erik the Outgolfer
Saved 1 byte thanks to Riley