Or you can get almost exactly 11 with one 9 only by applying a long series of ! and sqrt functions, as long as you accept factorials of non-integer values.

I.E where x! is gamma(x+1) for non integer values of x.

In this case the string...

s!!ss!ss!s!s!s!s!s!ss!ss!!ss!s!s!ss!s!s!s!ss!!s!ss!s!s!s!!sss!!s!ss!s!s!!ss!s!ss

...applied to a single 9 will equal 11.00004619 (where, reading from left to right, you apply 's' (sqrt) once then '!' (factorial) twice, then 's' (sqrt) twice, then ! once, then sqrt twice etc....

There will be some finite combination of ! and sqrt functions that will get you arbitrarily close to 11.

You can get almost exactly 11 with one 9 only by applying a long series of ! and sqrt functions, as long as you accept factorials of non-integer values.

I.E where x! is gamma(x+1) for non integer values of x.

In this case the string...

s!!ss!ss!s!s!s!s!s!ss!ss!!ss!s!s!ss!s!s!s!ss!!s!ss!s!s!s!!sss!!s!ss!s!s!!ss!s!ss

...applied to a single 9 will equal 11.00004619 (where, reading from left to right, you apply 's' (sqrt) once then '!' (factorial) twice, then 's' (sqrt) twice, then ! once, then sqrt twice etc....

There will be some finite combination of ! and sqrt functions that will get you arbitrarily close to 11. For sufficiently large N, you should be able to get ≥N decimal places accuracy by 'applying' a string of s's and !'s kN characters long (where k is an unknown constant which I estimate to be less than about 50).

A similar argument for the other numbers that need 2 9's would allow you to generate all 12 numbers with only 12 9's.