# Haskell, 100 bytes, score ≈ 10↑↑65503

    f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)q[q|_<-r,_<-r]f
    main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. `pred ':'` is equal to `'9'`.

I'll be severely impressed if someone can tell me what my score is.
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### Arguably more readable version in C-like psuedocode:
    myFunc = (n => f => {
      if (n == 0) return f

      q = myFunc (n-1)
      r = q(q(f)) ("a")

      fa = q
      for (i=0; i < r.length^2; i++)
        fa = (func => q (fa (func)))

      return fa (f)
    })

    print myFunc(65502)(str => str + str)("9")