There’s nothing stopping you from defining such a number to exist. The consequences of what we’ll call k are all sorts of cool: all numbers which are plus or minus a natural number are bigger than any natural number. Subtract all you want, you’ll never get a natural number out of the results. -k is smaller than any natural number or integer, so all valid k-numbers bracket around all other numbers. A simple sign flip causes them to sail all the way over every natural number or integer.

If you use real numbers instead, then the inverse of k is infinitely smaller than the real number closest to zero, yet between that mythical real number and k is an infinite number of multiples of k. Nor is there anything stopping you from defining a k’ which is larger than all of the numbers you can make with k, nor a k”, nor a …

Math is damn amazing, at times.