True story!

Finn is driving to his friend's house when he realizes that he's forgotten his phone, which has her exact address. Though he's able to find the street, he doesn't want to knock on the wrong door. The only thing he remembers about her house number is that it's the smallest positive integer such that the sum of the divisors of the sum of the divisors of the sum of the divisors of the house number is $840.$

What's her house number?

In other words, if $f(n)$ gives the sum of the positive divisors of a number including that number, then solve for the smallest $n$ such that $f\big( f( {\small f(n)} )\big)=840.$ To clarify, for $n=12,$ $f(12)=1+2+3+4+6+12=28.$

Note: This problem originally stated that there is a unique solution, which was incorrect. It now asks for the smallest. If I could re-do it, I'd ask the question as the sum of the solutions.