Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.
For example, consider the causal LTI system whose (infinite) impulse response is
$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$
Taking the Z-transform, we obtain the following non-rational transfer function
$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$
when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.
Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.
Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation
$$y (n) - \frac 12 y (n-1) = x (n)$$
which requires only $1$ adder, $1$ multiplier and $1$ delay.
Exponentials — whether real or complex — have low complexity, i.e., they can be generated using few adders, multipliers and delays. Do note that discrete-time exponentials are eigenfunctions of the delay operator.