3. The cunning rabbit and the foolish fox agreed: if the fox passes the bridge in front of the rabbit's house every time, the rabbit will give the fox money, doubling the fox's money, but the fox has to pay the rabbit $2^{2019}$ cents as a toll each time it crosses the bridge. The fox was very happy to hear that its money would double every time it crossed the bridge, but after crossing the bridge 2019 times, it found that all its money had been given to the rabbit. How much money did the fox initially have?
3. $2^{2019}-1$.

Let the fox initially have $a_{0}$ cents, and after the $n$-th crossing, it has $a_{n}$ cents.
$$
\begin{array}{l}
\text { Then } a_{n}=2 a_{n-1}-2^{2019} \\
\Rightarrow a_{n}-2^{2019}=2\left(a_{n-1}-2^{2019}\right) .
\end{array}
$$

Thus, $\left\{a_{n}-2^{2019}\right\}$ is a geometric sequence with the first term $a_{0}-2^{2019}$ and common ratio 2, i.e.,
$$
a_{n}-2^{2019}=2^{n}\left(a_{0}-2^{2019}\right) \text {. }
$$

Given $a_{2019}=0$, we have
$$
\begin{array}{l}
0-2^{2019}=2^{2019}\left(a_{0}-2^{2019}\right) \\
\Rightarrow a_{0}=2^{2019}-1 .
\end{array}
$$