Batteries with no regulator - Why is this a bad idea?

The ATMega 168/328 chips both can run on 1.8v to 5.5v power. Theoretically speaking, you can hook one up to a pair of AA batteries and it'll chug along happily even as the voltage drops.

I don't really feel comfortable doing this. There's just something nagging at me that says this is a Bad Idea, and that you really should put a regulator in there.

What problems would such unregulated power cause? Why is it a bad idea?

The chip can only run at 8MHz at 3.3 volts or lower (as per the datasheet).

Yep, hence why all the 3.3v arduinos are running at 8MHz. But if it weren't an issue with something, I feel there'd be more cheap arduino boards that ran without regulators and just ran off of something like a straight 3.7v battery.

You can do that - people here do it all the time. It's quite common with barebones Arduinos, and I think one of the lilypad boards comes like that (since they want to power it off lithium coin cell. If you're running off the internal oscillator (which you probably are), be aware that the speed (and hence the board's perception of time) will vary with battery voltage; there's more information on this in the datasheet too. You also need to have the BOD fuses set correctly to keep it from going into BOR at a higher voltage than you want it to.

Re: clock speed - there's a graph in the datasheet of speed vs voltage. Be sure to choose a speed that will work in the expected operating voltage range - though it's been widely reported that the '328p at room temperature can go faster than spec'ed; apparently it usually works at 16mhz at 3.3v (it's spec'ed for only 10 or 12 at that voltage, I think)

DrAzzy:
If you're running off the internal oscillator (which you probably are), be aware that the speed (and hence the board's perception of time) will vary with battery voltage; there's more information on this in the datasheet too.

This is probably the primary reason why - Having your well-timed things slowly become not-so-well-timed would end up being an issue, depending on what you're doing.

You could use the Arduino / Atemga chip to measure the battery voltage and have it warn you or stop doing stuff if it falls below some threshold.

...R

AndonRT:
This is probably the primary reason why - Having your well-timed things slowly become not-so-well-timed would end up being an issue, depending on what you're doing.

Do not conflate. @DrAzzy raised an issue that is specific to the internal calibrated oscillator. The issue of clock accuracy is independent of how well the processor runs over a voltage range. If an accurate clock is important then provide one.

Why is it a bad idea?

Batteries are an excellent source of electricity. Running the processor directly from batteries is more efficient. A pair of AA batteries provides the right voltage level. What part of that is bad?

AndonRT:
But if it weren't an issue with something, I feel there'd be more cheap arduino boards that ran without regulators and just ran off of something like a straight 3.7v battery.

Buy an 8 MHz Pro Mini from EBay. Upload sketch. Disconnect from computer. Connect 3.7V battery to the VCC pin. Enjoy.

There are cheap Arduino boards available that can run without regulators.