It depends on what you are trying to achieve.
If you need exact figures for the maximum tidal height each day/year at a given location, there are no simple analytical solutions due to the complex nature of tides.
If you are interested in ballpark figures (which are good enough for most recreational, but also some engineering applications), the maximum annual ...
You are looking for an easy solution to the local maxima of
$$\sum_i a_i \sin(\theta_i+\omega_i t)$$
This means solving for solutions to
$$\sum_i a_i \omega_i \cos(\theta_i+\omega_i t) = 0$$
This is a transcendental equation with multiple terms with different frequencies. While there are many ways to find such maxima that vary in computational complexity and ...