Real Analysis – What are some of the most unexpected constants in mathematics?

I am preparing work for students in computer mathematics. As a task, I want to entrust them with the task of writing programs to compute mathematical constants, which is not so easy to calculate except by brute force. I've thought of elusive, strange or unexpected constants. Of course, this question will not be well defined unless we define what "unexpected" means. But before defining, here is perhaps the strangest constant we know, but there is no way to calculate it and, in the future, no one will ever find a way to calculate it.

Constant chaitin $ Omega_F $ or stop probability is a real number that, informally, represents the probability of stopping a randomly built program. Each probability of stopping is a normal, transcendental real number that is not calculable, which means that there is no algorithm to calculate its numbers. In fact, each probability of stopping is Martin-Löf random, which means that there is not even an algorithm to reliably guess its numbers.

Definition: We define as a strange or unexpected constant if it can not be expressed in a close way
form in terms of other well known mathematical constants other than
its own definition.

According to this definition, popular constants such as $ pi, e, gamma $ etc. are not unexpected because we have many formulas that use integers to express these constants.

Examples of an unexpected mathematical.

  1. The constant of Mill, defined as the smallest positive real number $ A $ such as the whole part of $ A ^ {3 ^ {n}} $ is a prime number for an always positive integer $ n $.
  2. Brun's constant $ B $ which is the sum of the inverse of twin prime numbers. I am not aware of any closed representation of $ B $ other than its own definition.