How likely is it that unpackaged, assorted batteries stored together will actually explode?

They most likely would not explode, but simply split open and ooze acid all over the place.

It’s been my experience that a battery left in a device for long enough will leak and generally get nasty. It’s not been a problem with uninstalled batteries.
If you’ve never had an incident, then it’s most likely a low possibility. Might even simply be lawyer repellent, based on a possibility.

I look forward to @jerv’s answer.

Unlikely. Alkaline batteries don’t do that.

Lead-acid batteries like the one in your car have enough power to cause issues if they short out, and also give off potentially explosive vapors if charged in a place without adequate ventilation. Lithium-ion batteries also have enough power to get hot enough to start a fire under certain conditions, usually involving improper charging or severe physical damage to the battery.

But regular alkaline batteries don’t have much juice, and are highly unlikely to randomly hit in a way that forms a complete circuit so they won’t short out. The worst they might do is rust and leak electrolyte, but they won’t explode.

@jerv is right. Also, it is hard to short the + and – of common batteries: AAA, AA, C, D since the contacts are so far apart. Not so for button batteries and 9V.

@RocketGuy I don’t consider 9-volts or button cells to be “regular”.

Slim to none. There has to be a circuit for there to be heat and it has to be along-term circuit for the heat to get enough to explode.

I suppose it is theoretically possible (like a thousand monkeys typing the declaration of independence), but it sure isn’t likely.

As a caution, these are a couple of links talking about battery fires.

http://themotherlist.com/fire-safety-video-must-watch/

http://www.nh.gov/safety/divisions/firesafety/news/20120808-9-volt-batteries.html

@Dancingmomo – the videos are entertaining. But the laws of physics still hold.