I used to do this for a co-worker and me. We'd use three AeroPress spoonfuls of coffee (which is a lot) to make two cups. I think the theory was that the saturation of that much coffee grounds in the small amount of water that took up what little room was left in the AeroPress chamber, was not efficient at all. So we'd brew the same grounds twice. This made for more bitter coffee, because the coffee was over extracted, and it pulled the tannins into solution.
We opted for simply letting the coffee brew longer and increasing pressure during the press to force extract more out of the grounds. This yielded a much stronger concentrate that we'd then dilute to normal coffee/water proportions. This created a delicious and full cup of coffee every time.
I think for your case, you'd be brewing hours apart with the same grounds to save money, and I think you totally could. The issue I see is that most caffeine in coffee is extracted within seconds of brew time, so your second brew with those grounds may not yield a nicely drugged cup of joe. So not only will the taste be bitter, but it may also put you to sleep.