Do operators exist in between the common ones?

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I might be in over my head asking this question as I am only a rising sophomore in high school having just finished algebra 2, so I probably won't understand any complicated math concepts, but I'll try my best. Anyway, Recently I've been thinking about how addition is repeated incrementation, multiplication is repeated addition, exponentiation is rep. multiplication, tetration is rep. exponentiation, so on and so forth, and I was curious if it was possible for there to be operators in between. For example, if we used the notation +x to represent our operator, x=1 being addition, x=2 being multiplication and so on, what would +1.5 be? What would be the properties of it?