From what I remember, we believe that the kings were exempt from the training. This seems surprising because one of the kings was expected to lead and fight in military expeditions. If one of the kings failed the training as so many Spartans did, it would lead to ridicule. The Spartans ridiculed those who failed in the training, and it would be unthinkable for the law abiding Spartans to ridicule one of their lawfully instated kings. This ridicule would upset the order of the Sparta.

The heir apparent was exempt from the agoge and would undergo a separate training. However, other children of the two kings would go through the agoge and if someone other than the heir apparent ascended to the throne they would have gone through the agoge.

Source: Paul Cartledge's Spartan Reflections

There's an excerpt in Plutarch (I can't remember where or speaking on who) that mentions a Spartan king that was great because he took part in the Agoge (Spartan training). SO if he is to be believed, then yes, kings were exempt, but did have the option.