Appending String Lengths

JavaScript (ES6), 32 bytes

f=(s,n=0)=>(s+n)[n]?f(s,n+1):s+n

How it works

f = (s, n = 0) =>   // given a string 's' and starting with n = 0:
  (s + n)[n] ?      // if the Nth character of (s + n) exists:
    f(s, n + 1)     //   try again with n + 1
  :                 // else
    s + n           //   return s + n

Starting with N=0, we test the Nth character (0-based) of the string made of the concatenation of the original input string and the decimal representation of N. We increment N until this character doesn't exist anymore.

Example:

N =  0 : abcdefghi0
         ^
N =  1 : abcdefghi1
          ^
N =  2 : abcdefghi2
           ^
...
N =  8 : abcdefghi8
                 ^
N =  9 : abcdefghi9
                  ^
N = 10 : abcdefghi10
                   ^
N = 11 : abcdefghi11    -> success
                    ^

Test cases

f=(s,n=0)=>(s+n)[n]?f(s,n+1):s+n

console.log(f("aaa"));       // -> aaa4
console.log(f(""));          // -> 1
console.log(f("aaaaaaaa"));  // -> aaaaaaaa9
console.log(f("aaaaaaaaa")); // -> aaaaaaaaa11
console.log(f("a1"));        // -> a13

LaTeX, 108/171

\newcounter{c}\def\s#1#2]{\stepcounter{c}\def\t{#2}\ifx\empty\t\arabic{c}\else#1\s#2]\fi}\def\q[#1]{\s#10]}

\q[] //1

JavaScript (ES6), 37 bytes

f=(s,t=s,u=s+t.length)=>t==u?t:f(s,u)

<input oninput=o.textContent=f(this.value)><pre id=o>