Please check my answer

There was one student still taking the exam when the bell rang.

Student is the subject and exam is the simple subject.

You're half right.

Student is the simple subject. What was the student doing?

taking the exam

Yes. Which one of those is the verb?

Also note the helping verb.

https://www.englishclub.com/grammar/verbs-classification-helping.htm

there is the verb.

helping verb is was
am I close?

I don't see any action in helping. Plus I already told you that one of three words is the verb!

Yes, "was" is the helping verb.

Taking is the verb

Yes. The verb is WAS TAKING.

Ok I think I got it. The complete subject is taking the exam

Huh??????

Actually, in the sentence "There was one student still taking the exam when the bell rang," the subject is "one student."

To identify the subject of a sentence, you can ask the question "Who or what is doing the action?" In this case, the action is "taking the exam," and the answer to the question is "one student."

Therefore, the subject of the sentence is "one student," not "exam." The word "exam" is the object of the preposition "taking," and in this case, it functions as the direct object of the action.