$ x +^{-1} y $ seems like a good notation in that $x^{-1}+^{-1} y^{-1}=(x+y)^{-1} $, and
$$
\frac{1}{x}\frac{1}{+}\frac{1}{y} = \frac{1}{x+y}
$$
(is this the "Freshman's Dream" in another incarnation?)