Let the 9 numbers from smallest to largest be
represented by the letters from "a" through "i", where

a < b < c < d < e < f < g < h < i

They have a mean of 9, so

a + b + c + d + e + f + g + h + i
—————————————————————————————————  = 9
              9   

or, multiplying both sides by 9

a + b + c + d + e + f + g + h + i = 81

The object is to make "i" as large as possible.

"i" will be largest when "a" through "h" are smallest

Since the median is 9, e=9, and we have:

a < b < c < d < 9 < f < g < h < i

To choose "a" through "h" as small as possible, we have

a=1, b=2, c=3, d=4, e=9, f=10, g=11, h=12

Then 

a + b + c + d + e +  f +  g +  h + i = 81

becomes

1 + 2 + 3 + 4 + 9 + 10 + 11 + 12 + i = 81

Solving for i:

                              52 + i = 81

                                   i = 29

That's the answer, 29.

This is the case when the 9 numbers are

a=1, b=2, c=3, d=4, e=9, f=10, g=11, h=12, i=29

Edwin