The digits of $2^{1000}$ are added together; the digits of that answer are added together; and so on until a single digit is left. What is that digit?

Adding the digits of a number never changes the remainder it leaves on division by 9, so the final single digit is simply $2^{1000}$ reduced modulo 9. The powers of 2 leave remainders 2, 4, 8, 7, 5, 1 and then repeat every 6 steps. Since $1000=6\times 166+4$, the answer is the fourth entry of that cycle, namely 7. The trap 6 uses the units-digit cycle 2, 4, 8, 6 of length 4, which is the wrong cycle for this job. Answering 4 reports 1000 reduced modulo 6 rather than the remainder that position points to, and 9 assumes any huge number must collapse to 9.