Learn Turkish: Gezi nerede bitecek? - Where will the trip end?

The verb root is bit- (to end), which ends in a consonant (t). You only need the buffer letter y (making it -yecek or -yacak) when the verb root ends in a vowel, like in okuyacak (will read). Since bit- ends in a consonant, we just attach -ecek directly according to e-type vowel harmony.

Doesn't a t at the end of a verb change to d when we add a vowel? Why does it stay t in bitecek?

Great catch! While the t to d change (consonant mutation) happens in a few specific verbs like git- (to go) becoming gidecek, most single-syllable verbs actually keep their original consonant. The verb bit- is one of those standard verbs, so its t does not change when we add the future suffix.

We are asking a question, so why isn't there a question particle like mi at the end of the sentence?

In Turkish, if a sentence already has a specific question word like nerede (where), kim (who), or ne (what), you don't use the yes/no question particle (mi / mı). You only use mi when you want a yes or no answer, such as in Gezi bitecek mi? (Will the trip end?).