Where are you, Heather?You asked about palm trees. I don't know of any palm trees which are reliable outside here in the greater Washington, D.C. area. But the trunkless sorts often persist indefinitely.
Here on the east coast, hystrix has the best survival record that I'm aware of. Some of the trunkless Sabal might be even hardier.
I've had Rhapidophyllum outside for decades. I've also had Trachycarpus fortunei outside for decades, but it is in a very protected place and has no above-ground trunk to speak of.
Jim McKenneyMontgomery County, Maryland, USA, USDA zone 7, not exactly palm country.

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