Second attempt

We can compute the logistic map using finite, but arbitrarily large binary expansions. Notice that this is different from multiple precision, as the size of a multiple-precision number is fixed in advance. This is

1.

reliable, but

2.

unfeasible in terms of time and space consumption.

For n > 10, the computation of fⁿ(x₀) takes more than I was patient to wait for. Thus, this is even worse than rational arithmetic. It is easy to see why this is the case. If x has a binary expansion with n digits, then 1-x has often n digits as well. But if x and y have m and n digits then xy has mndigits. Thus, fⁿ(x₀) has approximately 2ⁿ m digits if x₀has m digits.