The long, long hours on the trail gave Doug a chance ----------

Concept: The noun "chance" (meaning an opportunity) is conventionally completed by a to-infinitive phrase — "a chance to do something" — not a gerund introduced by "for" or "in". Separately, "get to know somebody" is a fixed phrasal idiom meaning to become acquainted with someone, which is different from the causative pattern "get somebody to do something" (meaning to make or persuade someone to do something).

Application: The blank must (1) form a to-infinitive after "chance", and (2) express Doug spending his solitary trail hours becoming acquainted with himself, which calls for the idiom "get to know" rather than a causative construction. "to get to know himself better" satisfies both requirements: it is a to-infinitive, it uses "get to know" with "himself" correctly as its object, and it keeps the natural word order verb + object + adverb ("know himself better").

Cross-check — why each other completion breaks the pattern:

• "to get himself to know better" keeps the to-infinitive but restructures it as the causative pattern "get + object + to-infinitive", giving the standalone idiom "know better" (to have more sense or judgment) rather than expressing him becoming better acquainted with himself.

• "for getting to know himself better" replaces the required to-infinitive with a "for + gerund" phrase, which is not the standard collocation after "chance".

• "in getting better to know himself" uses an "in + gerund" phrase, also not accepted after "chance", and additionally reverses the correct word order by placing "better" before "to know himself" instead of after it.

Result: Only the to-infinitive phrase built on the idiom "get to know", in natural word order, completes the sentence correctly, confirming that completion as the answer.