You can pour hot water over used coffee grounds and get a brown liquid out the other end. Whether you can call that liquid coffee is a different question.
What happens chemically
The first brew extracts most of the desirable compounds: sugars, oils, balanced acids, and aromatic flavors. By the time the first cup is done, roughly 18 to 22 percent of the coffee mass has dissolved, which is the sweet spot for taste. What remains is largely cellulose and the harsher, less-soluble compounds that did not come out the first time.
A second brew pulls those harsher compounds without the sweetness to balance them. The result is bitter, thin, and astringent.
The exception: cascara and steep extensions
Some drinkers extend the steep on a single brew rather than doing two. With a French press, leaving the grounds in for 6 to 8 minutes instead of 4 will extract more, though you risk bitterness. This is different from a true second brew because the grounds are still doing their first-pass extraction.
Practical alternatives
If you are trying to stretch your coffee budget, the better moves are to grind slightly finer (extracts more from the same amount), brew at a stronger ratio (1:14 instead of 1:16), or buy in slightly larger quantities to capture small bulk discounts.
If you genuinely want to reuse grounds, they have a much better second life as garden compost, deodorizer for the fridge, or scrub for greasy pans. They are excellent at all three. Just not at making coffee.