Why is Euler's identity considered so miraculous and beautiful? 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.
 A: 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.
A: 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'
A: 
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:


*

*Has anyone talked themselves into understanding Euler's identity a bit?

*Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

*Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

*Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $\pi$ are related?

*What could the ratio of two sides of a triangle possibly have to do with exponential functions?

*Factorial in power series; intuitive/combinatorial interpretation?

*How fundamental is Euler's identity, really?
A: When we work in degrees, Euler's identity will take the form of
$e^{i 180^\circ}+1=0$
which has nothing special. The units we use to measure angles are not great from a mathematical point of view and are chosen arbitrarily. The pi can be replaced by any number in this famous identity, which means it shouldn't even be ranked among the top 100 most beautiful relationships. This, of course, does not diminish Euler's personality at all.
Scientific calculators help perpetuate the artificial beauty of identity, as when we choose to work in degrees, the powers of numbers containing the imaginary unit are calculated in rad.
