Given its success in other renewable energy domains, such a solar and wind, extremum-seeking control (ESC) would seem to be a promising candidate for the wave energy energy-maximising control domain. There are many advantages to ESC, principal among them the lack of a need for a mathematical system model, with effective wave energy hydrodynamic modelling being particularly challenging. However, a number of fundamental limitations of ESC for wave energy converters (WECs) can be found, with the panchromatic (stochastic) nature of the wave excitation a key issue, resulting in potentially long performance function evaluation times. This, combined with the desire to perform intra-wave control adjustment, creates a natural tension in the solution of the wave energy ESC problem. Motivated by this, we examine, in this paper, the fundamental opportunities and pitfalls of ESC for the wave energy control problem, providing a concise definition of WEC ESC and its characteristics. In particular, we investigate the intrinsic limitations behind ESC for the WEC control problem, and provide a potential set of future directions aiming at alleviating such disadvantages and directly contributing towards the development of efficient model-free control systems for wave energy devices