Can the original Gameboy be played with an AC/DC adapter?

As Wikipedia clearly states, the original Gameboy can be used with an AC/DC adapter:

The Game Boy also contains optional input and/or output connectors. On the left side of the system is an external 3.5mm × 1.35mm DC power supply jack that allows users to use an external rechargeable battery pack or AC adapter (sold separately) instead of four AA batteries.

Here's an image of the official power supply sold for the system:

image source

Yes, the Gameboy has a DC jack:

However, it is one with a built-in switch to switch between the batteries and the DC input. This switch isn't perfect and the Gameboy has very small (nonexistent?) supply buffer caps, so plugging/unplugging the adapter during a game usually crashes or resets the game.

It is not used to recharge the batteries. (The Gameboy can be powered from non-rechargeable batteries.)

I can confirm that I had a Gameboy (Pocket, but my brother had a normal one) and an adaptor for it.

I think you were technically meant to use it on a Gameboy without any batteries in, but we frequently used it with batteries and it did charge them to an extent, though I couldn't recommend you did so.
I believe it could also be used with proper rechargeable batteries to recharge them too.