Rise, sequence, rise
Pyth, 31 30 29 bytes

e+0f.x!sgM.:iVczdT2ZosN^STlcz


1 byte thanks to @Jakube.

Demonstration. Test Harness.

Input is given on STDIN, space separated. If newline separated input is allowed, I can shorten the program by 2 bytes.

Explanation:

e+0f.x!sgM.:iVczdT2ZosN^STlcz
                                  Implicit: z = input(), T = 10, Z = 0, d = ' '
                        ST        [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
                          lcz     len(z.split())
                       ^          All combinations w/replacement of that length.
                    osN           Order by increasing sum.
   f                              Filter on
              czd                 z.split(' ')
            iV   T                Vectorize the "Convert to base" operation over 
                                  the integers as strings and the base sequence.
          .:      2               Take length 2 subsequences.
        gM                        Map the &gt;= operation over them.
      !s                          Sum and logically negate.
    .x             Z              If that throws an error, returns 0 (e.g. reject)
 +0                               Prepend a 0, in case no sequences are found.
e                                 Take the end of the list.


Including 1 in the list of possible bases is safe because i, which uses Python's int builtin, doesn't allow 1 as a base, and therefore always throws an error, which is caught and filtered out.