Should one bring a power strip when travelling from the U.S. to Europe so that one won't have to get more than one power adapter?

Rather than a whole power strip which is very bulky, consider bringing two adaptors for the area you are visiting, and two three-way outlet adapters - they are very compact solid blocks that plug in and offer outlets on three sides. Between them in a hotel you have six three-prong outlets you can make use of.

They are also great to have in carry-on for airports, if someone is already plugged into an outlet you can ask if it's OK to unplug them for just a second to open up two more outlets you can use.

That could be a good idea. Be aware that in general Europe has 220v whereas the US has 110v. Most modern laptops support both, but make sure all of your appliances do too.

You might also want to look at this related question, one of the answers there mentions a webshop in the us where this very handy universal powerstrip is for sale.

The Wonpro Nano is a roughly 1"x1"x1.5" cube and you can plug two things into. The ultra small Mogics adapter/power strip is promised for next month. These are not grounded solutins, however and you should only plug ungrounded plugs into them. Finding universal adapters for grounded plugs is not easy. As far as I am aware, only Skross has three pole universal adapters (the Tumi adapter is a relabeled version made by Skross, check the bottom for the Skross logo/label) and then you can use a Voltage Valet PS2 power strip which, being a travel power strip is rather compact.

Be extremely wary of Monster, they are a tax evading patent troll and as such, boycott worthy.