For n data points, there is a unique polynomial of degree n−1 that passes through the points; one way to get it is the Lagrange polynomial.
Another class of interpolating functions is the thin plate spline, which is typically used for 2D image warping, but can easily be applied to 1D interpolation as well, which is a strictly simpler problem. The Wikipedia article on polyharmonic splines, of which the thin plate spline is a special case, has examples of other infinitely differentiable splines.
I believe that in principle, you can choose any basis function whose translations are linearly independent, and find an interpolating function as a sum of translations and scalings of the basis. The above are essentially examples of particular basis functions people have used because of their other nice properties.
Edit: Sorry, thin plate splines and other examples in the article, apart from the Gaussian, are infinitely differentiable almost everywhere. However, if you really want infinite differentiability everywhere, you can use any infinitely differentiable function as basis instead, such as the Gaussian, or $x^2/\sqrt{x^2+1}$.