What's the space-complexity of Newton-Raphson? I think it reduces to the space-complexity of storing the inverse hessian matrix.
No, it reduces to the space complexity of solving a linear system using the Hessian matrix.
As with all situations where you need to solve a linear system, computing and storing the inverse explicitly is a bad idea for any problem larger than about 4 by 4. The inverse of a matrix is usually not as well-conditioned as the matrix itself, the inverse of a sparse matrix is typically not sparse... all the usual reasons apply.