Knuth's Up-Arrow Notation is a mathematical notation introduced by Donald Knuth to express very large integers. It uses arrows to represent repeated exponentiation, which allows for a compact way to describe numbers that grow extremely fast. For example, a single arrow (a ↑ b) represents a^b, while two arrows (a ↑↑ b) denote a raised to itself b times. The notation can be extended with more arrows for even larger numbers. For instance, three arrows (a ↑↑↑ b) represent a power tower of height b, where a is the base. This system provides a clear framework for discussing large values in mathematics.