Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily large, but also that no matter how far you go, you will always eventually find [small gaps](https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/).

What are other examples of objects in mathematics that crop up without any discernible patters, but are totally unconnected to prime numbers?