##JavaScript (ES6), 37 34 bytes
f=(n,s='1')=>--n?s+f(n,++s[0]+s):s
In JavaScript, strings are immutable. Therefore, it's impossible to alter the content of the Nth character of a string s by assigning a new value to s[N].
However, the expression ++s[N] does evaluate as one would expect, even if the string remains unchanged. For instance:
++"1"[0] // equals 2
And by extension:
s = "21"
++s[0] + s // equals "321"
###Demo
f=(n,s='1')=>--n?s+f(n,++s[0]+s):s
console.log(f(6))