Can MacBooks run without a battery?

I have a MacBook Pro (2008). It is a secondary computer I rarely use, so it was easy to last that long with it.

Such a long lasting service turned me into an adept of Apple quality. I am looking forward to buy a new one except... nowadays the battery is harder to remove.

My concern is not "a replaceable battery", it is more of "a removable battery". As soon as the battery dies, I would like to keep using the laptop with a power cord. However I've seen all my batteries swelling and/or getting extremely hot. I might be paranoid, but in those cases I just wanted to remove the battery and keep using a good enough laptop.

Is this possible with new MacBooks? Can they work without a battery attached? Or can their batteries be ignored once they expire?

It is well-known that any MacBook with battery removed will run at significantly lower clock speed, typically around 1 GHz. The reason is that the power supply is good enough to supply plenty of power for the average usage, but not for bursts. If your hard drive spins up while the CD drive spins up while all cores and the GPU work full blast and you are charging two iPhones then the power supply won't have enough power for that, but it can take extra power from the battery.

With no battery, the MacBook will turn its power consumption down so that it can guarantee to always have enough power, because otherwise you would have instant crashes at the wrong times.

Another reason to not remove the battery is that dust will have a chance to get deeper inside your MacBook that would otherwise be blocked by the battery. So unless your battery is swollen, leave it in, even if it only has ten minutes charge left.

MacBook Pro Mid 2015 cannot be run without battery at all.

I can confirm that a Macbook Pro Retina 2015 (15") WILL NOT boot correctly without a battery installed. I was rebuilding a system that had damaged batteries. Once I pulled the old battery pack you could show boot disk screen, perform hardware check etc, but cannot boot to OS. You just see the Apple boot screen but will be stuck in a reset loop. Also fans were on 100%.

Started working fine once new battery pack was installed.

Well, even tho the battery of the new MacBook pro's are glued with adhesive, they are removable, just harder. Plenty of youtube tutorials explain how to do it. So yeah buy yourself a new battery and replace it and do not run the laptop without a battery.