How can I splice strings of patio lights together?
"Holiday" lighting is covered in the code and required to be listed by Article 410.160. It is considered temporary lighting and therefore you could argue that it could be spliced without a junction box.

The code puts a 90 limit on holiday lighting [590.3(B)] but there is a holiday at least every 90 days and you could claim that you are celebrating a holiday practically every day of the year.

If you don't mind violating the listing of the assembly, and the code: Solder them together and use shrink tubing to cover the splice (make sure you put the shrink tube on one side before you solder the splice.) It may not be pretty but if it is well insulated it will be as safe as the plugs and probably more waterproof. Keep the 3 amp fuse on the end of it for added safety.

OR as bats suggested get a roll of LED rope light terminated with an approved plug.

Happy Thursday!