The sentence should really include the missing preposition of :
The king had many occasions of experiencing the power of the hermit's talisman.
This would make it equivalent to any number of similar constructions:
The king had many occasions of good health.
The king had many occasions of state.
Strangely, it seems to be understandable without the preposition even though the other constructions aren't.
The difference between it and the other example you gave is that this example is discussing a series of things, whereas the other example is discussing a single, and continuous, thing. However, the general construction is the same—and in the other example, the preposition used is for.
Having said that, even though it's grammatical, it's not entirely idiomatic and it would commonly be phrased differently:
The king had many occasions to experience the power of the hermit's talisman.
To address a point in another answer, although verb tenses can be legitimately mixed in some sentences, the issue is moot here because this particular sentence only has a single verb: to have.
Here, experiencing is not a verb, it's a gerund—or an -ing-formed verb that is acting as a noun. As such, it is without a tense. (He was smoking, he is smoking, he will be smoking.)