You and I would be great friends in real life entropie.
Just to add onto the point...
One can interpret what is happening is the limit of model for "growth" through compounding split over n periods.
If n=2, (1.5)^2=2.25
If n=3, (1.333...)^3= 2.370370...
If n=4, (1.25)^4=2.44140625 (and change)
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So you can interpret e as the result instantaneous compounding done by splitting the growth into infinitely small compoundings split over infinite number of times.