"Other types of batteries may burst causing injury to persons and damage," cautions your typical battery charger. It's a pretty serious warning that we unquestioningly follow. Disposable alkaline batteries and rechargeable nickel-metal hydride cells (NiMH) use different chemicals, so it makes sense that we can't just interchange our batteries. Seems straightforward enough, right?
Scientific American article: "One of the necessary conditions for a battery to be rechargeable is that the underlying chemical changes that occur during an electrical discharge from the cell must be efficiently reversed when an opposite electrical potential is applied across the cell." It must then be able to sustain this process efficiently and safely over many charging cycles for a battery to be labeled rechargeable.
While your typical alkaline battery has a reversible chemical process it performs more poorly with each charge. "In the case of the nonrechargeable battery," writes Frank McLarnon in Scientific American, "when one attempts to recharge the battery by reversing the direction of electron current flow, at least one of the electrochemical oxidation-reduction reactions is not reversible." The result is a lossy charge. And this assumes that a buildup of hydrogen gas in the reverse process hasn't caused the battery to rupture.
Even if it works, is it worth it? Perhaps not. Watts Clever recommends that the batteries being recharged should have been manufactured at the same time and, because they use different chemical compositions, batteries of different brands should not be mixed. The lossy nature of the process might be frustrating as the life of the battery decreases unpredictably.
Its utility depends on how many alkaline batteries you have laying around to use with the $46 device. You're probably better off just gathering up your old AAs, taking them to be recycled and replacing them with NiMH rechargeables.