What you describe is closely related to what is called recreational mathematics. One could make the case that games, puzzles, and doing things just for fun are just as much a part of "real life" as anything else, but when people say "practical applications from real life," they usually tacitly exclude recreation.
It is not easy to come up with mathematical problems that are neither "too hard" nor "too easy." Recreational mathematics has a tendency to fall into one of these two categories, and hence rarely develops into what we would call a "branch of math." For example, the question of the existence of Lychrel numbers was probably first "randomly" asked out of curiosity by someone with no practical application in mind. Although there has been quite a bit of effort devoted to Lychrel numbers, the subject is "too hard" in the sense that it seems difficult if not impossible to make significant theoretical progress on it. So Lychrel numbers are not what one would call a "branch of math."
The surreal numbers are perhaps an exception in that they originated in recreational study of the game of Go, but have turned out to be a rich subject of mathematical study. Origami has its roots in recreation, or at least in art, and its mathematical study has turned out to be surprisingly rich, with applications in engineering.
EDIT: Another interesting case study is graph theory. Its origins can be traced to the seven bridges of Königsberg and the four-color problem—questions that were driven primarily by curiosity rather than practical necessity. There is no doubt that graph theory is now a major branch of mathematics. However, one could argue that its development into a major branch of mathematics was driven by the rise of computers and the many applications of graph theory to computer science and electrical engineering.
It's also worth mentioning that even in well-established areas of mathematics, new subfields often arise not because of perceived applications, but because of questions that mathematicians find interesting to study. If you're looking for examples of areas of mathematics that were born from "curiosity-driven" rather than "applications-driven" reasons, then there are too many to list.