Annoying primes first of all, I'm no mathematician at all. I was just playing with prime numbers and ended up with this list:
2 = 2¹
3 = 3¹
5 = 5¹
7 = 3¹ + 2²
11 = 2¹ + 3²
13 = 13¹
17 = 5¹ + 2² + 2³
19 = 2¹ + 3² + 2³
23 = 11¹ + 2² + 2³
29 = 17¹ + 2² + 2³
31 = 19¹ + 2² + 2³
37 = 37¹
41 = 5¹ + 3² + 3³
43 = 7¹ + 3² + 3³
47 = 11¹ + 3² + 3³
53 = 17¹ + 3² + 3³
59 = 23¹ + 3² + 3³
61 = 61¹
67 = 7¹ + 2² + 2³ + 24 + 25
71 = 67¹ + 2²
73 = 61¹ + 2² + 2³
79 = 3¹ + 7² + 3³
83 = 79¹ + 2²
89 = 13¹ + 7² + 3³
97 = 13¹ + 3² + 3³ + 24 + 25
101 = 97¹ + 22
103 = 11¹ + 72 + 33 + 24
107 = 103¹ + 22
109 = 17¹ + 72 + 33 + 24
Edit2: As pointed out by @Charles, 29, 67 and 97 aren't annoying.
I was just playing by trying to write the primes following this form
a1 + b2 + c3 ...
(just one rule, there must be all the exponents, or else the number must be written by itself. So, I can't have a number written in the form a1 + b3 + c4)
Edit: There are actually two rules, I forgot to say that a, b, c... must be primes.
Some of the primes, the bold ones, can only be written with exponent 1. I called then annoying primes, in lack of a better name.
Is it some property of these primes? Is there a formula to express this?

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.
