Safe to plug a travel adapter into a power strip?
Is that too many layers of things where it can be a problem safety wise?

The number of &quot;layers&quot; isn't a huge concern, no. As long as each individual component is rated for the voltage and current that will be present, it should be fine.
That said, you might not want to use this approach as a permanent solution. The travel adapters are useful, but in my experience don't always fit perfectly. They can wobble, loosen, etc. At the very least you might wind up with exposed conductors where you don't want them, and I suppose in the worst-case scenario, over time you might get some electrical arcing.
If this is to be a long-term connection, consider replacing the cord on the lamp. Since the lamp itself is already compatible with dual voltage, replacing the cord shouldn't be a huge task. It depends on the design of the lamp, of course, but many lamps can be relatively easily dismantled to gain access to the cord's connections, allowing replacement.
Alternatively, it may be possible to find a power strip that has outlets compatible with both the North American standard and the European standard you're dealing with. For example, I find myself in Singapore from time to time (in-laws) and have purchased power strips there which have outlets that each can accept either the Type C plug that is sometimes used there (though Type G is more common) or the Type A/B plug that is used in North America.
You might be able to find a similar power strip for your purpose (i.e. one where each outlet is compatible with both plug styles), or one that has some plugs of one type and some of another.
Whether this is possible depends on what type of plug exactly you're dealing with. Your post is not specific about that, so I can't provide anything more exact than that.