When I divided $1$ by $9,899$, I got two-digit Fibonacci numbers also being carried: $0.0001010203050813213455904636\dots$
When I divided $1$ by $89$, I got one-digit Fibonacci numbers at the beginning: $0.0112359\dots$ (there was originally an eight, but there was carrying and it changed to a nine)
How does all of this happen? There is more than this, you know. What do you think is going on here?
This happens because the generating function of the Fibonacci numbers is $\frac x{1-x-x^2}$. At $x=0.01$, this is $\frac {100}{9899}$.