Since then you could presumably plug the power strip into the power adapter and then not have to get multiple power adapters?
Or is this unnecessary?
Traveling with a power-strip is an old trick for avoiding to carry multiple plug adaptors, but going from the US to Europe you need to be a little careful.
Electricity in the US is ~110 volts, whilst in much of the rest of the world it's 200-250 volts.
Although power boards/power strips are generally passive, and thus the number of volts should not have any impact, many of them do include various types of fuses or additional circuitry (eg, USB ports) that could potentially have issues with higher voltages. If you were to plug multiple high-current devices into a power board (eg, a hair dryer) it's also possible that you could draw more watts than the board is designed to support.
There are a few products that are specifically designed for travel, and designed to support both 110 and 240 volts, such as the Monster Outlets to Go range (Note: Amazon doesn't say it, but the manufacturer has confirmed that these items are designed for up to 250 volts)
Note that going the other way is far less of a problem - a power board designed for 250 Volts will work fine in the US - I have several where I have taken an Australia power strip, removed the Australian plus and put on a US plug so I don't even need an adapter to use it!
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.
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.