The Hyperoperation Sequence is a series of operations that extends beyond basic arithmetic. It starts with addition, then moves to multiplication, exponentiation, and continues to more complex operations like tetration and pentation. Each operation in the sequence builds on the previous one, allowing for increasingly large numbers to be represented and manipulated. For example, while addition combines numbers, multiplication can be seen as repeated addition. Similarly, exponentiation represents repeated multiplication. The sequence continues with operations like tetration, which is repeated exponentiation, and pentation, which is repeated tetration, showcasing the rapid growth of numbers as you progress through the sequence.