"J'aime regarder jouer les enfants."
No, your construction does not work. It means "j'aime regarder les enfants (en train de) jouer", which is the equivalent of the continuous "playing" form. closer to your proposition, you can have : "j'aime regarder quand les enfants jouent" (when the children are playing).
A number of verbs are possibly followed by another infinitive verb: j'aime faire, je regarde faire, je laisse faire, je veux faire, je peux faire, je dois faire... When two of them are combined, like in this example, you can have two infinitives one after another.
Note that "watch the children playing" uses "playing" in the gerund form, which is more rare in French and not used the same way as in English: "je regarde les enfants en écoutant de la musique" = I watch the children while listening to music.
There is another way to avoid a 3rd infinitive: "j'aime regarder les enfants qui jouent"
But the construction proposed indeed depends on the verbs, with the first one (the conjugated one) allowing the use of an infinitive (equivalent of "to Verb" or "Verb-ing" or "at/for/from Verb-ing))