I feel like this is a pretty difficult question to answer, more so than the other one but for philosophical reasons, not mathematical ones. I'm going to take a maybe unconventional and perhaps inflammatory approach. The problem is that math is inherently contrived and unintuitive by nature. It's a series of constructions that we've built up for thousands of years based on how we view the world. Could you really explain a function intuitively without appealing to really rigorous foundations like ordered pairs? You can't see or touch a function (though you do see their effects everywhere). However I would say that basic arithmetic with integers and rationals is the only real answer here.

I kind of follow Kronecker's thought process: God created the natural numbers and the rest is the work of man. I wouldn't appeal up God personally since I'm not religious but instead I would say that natural numbers are the closest things we could separate from the human condition. (Even this can be a matter of contention but then you get into solipsism and Descartes' issues with reality and that's not math anymore.)

Math is based upon a series of axioms and these axioms have been carefully chosen over time, much of which has been motivated by our own perception of the universe around us, even if we pure mathematicians might like to think it's not. We have no way of even conceiving of what math might be like to an alien civilization. Though it is hard to imagine an alien civilization not recognizing the role of integers and rationals in some way. As such, I maintain that math is unintuitive, since we can't even begin to guess what other civilizations might conclude, and it's just that some ideas are less unintuitive than others.