5. Let the function $f(n)$ be defined on the set of positive integers, for any positive integer $n$, we have $f(f(n))=4 n+3$, and for any non-negative integer $k$, we have
$$
f\left(2^{k}\right)=2^{k+1}+1 \text {. }
$$

Then $f(2303)=$
5.4607.

Notice
$$
\begin{array}{l}
2303=3+4 \times 3+4^{2} \times 3+4^{3} \times 3+4^{4} \times 2^{3} \text {. } \\
\text { And } f(4 n+3)=f(f(f(n)))=4 f(n)+3 \text {, then } \\
f(2303) \\
=3+4 f\left(3+4 \times 3+4^{2} \times 3+4^{3} \times 2^{3}\right) \\
=\cdots \\
=3+4 \times 3+4^{2} \times 3+4^{3} \times 3+4^{4} f\left(2^{3}\right) \\
=3+4 \times 3+4^{2} \times 3+4^{3} \times 3+4^{4}\left(2^{4}+1\right) \\
=2303+4^{4}\left(2^{4}+1-2^{3}\right)=4607 .
\end{array}
$$