[mikorym]

Sometimes I find these concrete investigations necessary for our brains to make peace with the unreasonable effectiveness of mathematics, as it's been called.

I would say one of the first great discoveries for a person is the exponential series (a real world examples: population growth). Another is the divergence of the harmonic series 1/n and convergence of 1/n^2 (my preferred real world example: pizza slices that converge to 1 pizza or diverge to infinitely many). E.g. give me 1/n slices for the rest of my life and I'll pay you $100 (-:

When travelling, I also have go-to experiments that I like doing (e.g., elementary proofs that the earth is round/spherical such as: great circles; N-E-S-W always at 90 degrees; shadow angles [Erastothenes]; seasons; etc.)

There are other things to investigate that are not really "proofs" or "combinatorial evidence", but equally interesting. One example is using music (esp. the piano) as a physical logarithm device. The music "sounds" additive but the frequencies are multiplicative.