You could look through the table of unicode input for alternatives. For example, \invv\2/3 which yields ʌ⅔.
Another idea would be something like this:
using SymPy: Sym, @vars
struct TwoThirds end
const _⅔ = TwoThirds()
Base.:(^)(x::Number, ::TwoThirds) = x^(2 / 3)
Base.:(^)(x::Sym, ::TwoThirds) = x^(2 // 3)
Now:
julia> a, b = @vars a b;
julia> 3.4 ^ _⅔
2.261097438656042
julia> a * b ^ _⅔
2/3
a⋅b
You could also define other methods for TwoThirds() like + or *.