A couple of years ago, I had a new idea - unfortunately, as I researched it, I realised that Donald Knuth (a man I respect) had had the same basic idea decades ago.
However, he did not appear to extend the idea as far as I've been thinking about it. The idea is this: The first operator in mathematics was to count up. Once you realise that repeated counting up is something that happens a lot, you make a shorthand for it, and call it addition. Once you realise that repeated addition is something that happens a lot, you make a new notation for it, and call it multiplication. Repeated multiplication gives the exponent (or power) operator. Repeated power can give something even newer.
The interesting thing is, that before the count operation was seen as something in and of itself, the set of numbers was basically limited to the numbers a brain can recognise instantly without counting - up to about 6. Once the count operator was recognised as something that can be applied indefinitely, the set of numbers expanded to include all the natural numbers. Once the addition operator was recognised as something in and of itself, the subtraction operator came along naturally - and the set of numbers is now all the integers, including the negative ones. Once the multiplication operator came into use, along with division, we got fractions and the rational numbers. Once the power operator came into use, and the roots were investigated, we got imaginary and complex numbers. At each stage, a new and unexpected set of numbers was revealed that was never even hinted at before. Each step has led to more expressive mathematics. If the pattern holds, investigating the inverse of Knuth's up-arrow notation should show an exciting whole new set of numbers. What x satisfies ((x ↑ 2) = (14 + 2i))? I think the answer to that question is actually a complex number. But what x satisfies (((2 - 5i) ↑ x) = -7)?
Yet more new numbers might arise from John Conway's extension to the up-arrow notation, his chained up-arrow notation. What might (3.5 → -1.7 → (5 + 2.2i)) evaluate to? And what about the inverses of the operator, e.g. what x satisfies ((1.84 → -5.2 → x) = 27)?