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For example, I could write this: $$(e+i\pi)^0=1$$ It has all the five constants and all the addition, multiplication, and exponentiation operators.

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    You achieve the same degree of beauty but with less economy of symbols. – 2017-02-24

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In your identity we can substitute the constants $e,i,\pi$ with any other number, because it is a proposition always true as a consequence of the axioms that define the operations ($x^0=1 \quad \forall x$).

The Euler identity is true only for the given numbers and expresses a property of these numbers.

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Your equation only tells you $x^0 = 1$ which is not especially interesting. The equation $e^{i \pi}+1=0$ could be considered neither trivial nor artificial and for the reasons you mention is considered, by some, to be beautiful. Beauty is of course subjective.

As a side note $e$ is not Euler's constant. See here

Edit: my side note was motivated by the original un-edited question having the tag 'eulers-constant'

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    About the side note, I think symbol $e$ comes from the name Euler, but yes Euler's constant is $\gamma$. +1 for the answer – 2017-02-24
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    The reason we use $e$ is because in Euler's original paper he had already used a-d and needed to call it something. – 2017-02-24
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It has all the five constants and all the addition, multiplication, and exponentiation operators.

Indeed it does ! Unfortunately, it is not particularly meaningful, as has already been pointed out. But why are Euler's identity and formula considered meaningful in the first place ?, you might legitimately ask me in return. To which I would like to respond by referring you to the following
seven posts: