Annoying primes

I can only find 6 annoying primes: 2, 3, 5, 13, 37, and 61.

You gave these and three other examples, but they don’t hold: $$29 = 17 + 2^2 + 2^3$$ $$67 = 7 + 2^2 + 2^3 + 2^4 + 2^5$$ $$97 = 13 + 3^2 + 3^3 + 2^4 + 2^5$$

By general density arguments one would expect:

Conjecture: There are only finitely many annoying primes.

I pose this problem which would strengthen my conjecture:

Open problem: Are there finitely many numbers which cannot be expressed with greatest exponent 2, 3, 4, or 5?

Sadly I do not have a computer here to check, perhaps someone else will do so and report back.