Multielectron atom has much more complex energy spectrum than hydrogen atom. As the electrons interact with each other, the hydrogenic energy levels get shifted, and much of the hydrogen-specific degeneracy, as well as degeneracy resulting from electrons mass&charge equality, is lifted. Moreover, since the electrons do interact with each other, we can't, strictly speaking, speak about single-particle states of electrons. This means that when we shine light on a multielectron atom, it can't excite one electron. It instead changes the configuration of the whole atom, and the final state is approximately what we would call an atom with single electron excited.
Now, as the energy levels are partially split by electron-electron interaction, there's a way$^\dagger$ to directly (i.e. via single photon) get to the state, which we approximately (i.e. in terms of the shell model) describe as twice-excited state. We just have to shine the appropriate frequency of light at the atom. This way we could even excite more than two electrons. Doing another way, we could shine light composed of two frequencies corresponding to transition from ground to singly excited state and from singly excited to doubly excited state. Then it'll also be possible for the atom to appear in doubly excited state, but this would be a sequential transition.
For some atoms though, doubly excited states are unstable in the sense that the corresponding energy is above ionization energy, so such states are likely to decay into ionized atom instead of directly going to ground state. This is true for e.g. doubly excited helium atom.
My intuition is that the atom would return to its ground state too quickly for this too happen
Note that transition from one state to another is not instant. The probability of occupying some state gradually changes with time, and if you shine the light driving transition to singly-excited state and mix the light driving transition from singly excited to doubly excited state into the same beam, you'll be able to "intercept" the atom's singly excited state and drive it to doubly excited one.
Also, I would suspect that if the atom could be excited twice, the already-excited electron would be excited (or ionised) a second time, not a new one.
Don't forget about quantization of energy and the fact that electron-electron interaction lifts much of the possible degeneracy. It's highly unlikely that energies of doubly excited atom and atom with single electron excited to higher level will coincide. Thus if you tune your light source well enough, you're unlikely to excite the already excited electron to higher level instead of getting doubly excited atom.
$^\dagger$ I may actually be wrong if it appears that such optical transition is forbidden for all atoms. If you find some evidence of this, please let me know.