Is this sentence a compound sentence, they spend most of their lives at sea, but they do come to shore for one purpose - to lay eggs.

This is a compound sentence because you can break it into two sentences.
Hope this helps

Yes, this sentence is a compound sentence. A compound sentence is a sentence that has two or more nouns or/and predicates joined with commas, semicolons, or conjunctions. In this is sentence, but is the conjunction. They is a predicate and sea and shore are nouns. Also you can break this sentence into two sentences: They spend most of their lives at sea. They come out to shore for one purpose- to lay eggs.

When the poor have cried, caesar bath wept/ ambition should have been made of sterner stuff