Well, there's the fake "proof by example" that square root is linear:
[sqrt(2)] + [sqrt(2)] = [sqrt(4)]
[sqrt(3)] + [sqrt(3)] = [sqrt(6)]
[sqrt(8)] + [sqrt(8)] = [sqrt(16)]
This works the same way as your example, by using the greatest integer operation to change the apparent value to something trivial.
Let I be the iteration operator, so that I(o,n) iterates the operation o n times. For an arbitrary ring G and a in G, let a+ be left addition by a and a* be left multiplication by a.
Theorem: For the real number field, there is a unique element a for which for which the additive and multiplicative operations coincide, i.e., I(a+,1)(a) = I(a*,1)(a).
2+2 = 4 = 2*2.
I just made this last one up on the spot. Hope it's amusing.