A compound-complex sentence is made from two independent clauses and one or more dependent clauses.   Compound-complex sentences are very common in English, but one mistake that students often make is to try to write them without having mastered the simple sentences, compound sentences, and complex sentences first.

examples of compound-complex sentences

“Although I want to take both the dogs on the hike, I think Maxi is too old, and I need to find someone to house-sit.”
independent clause: “I think Maxi is too old”
independent clause: “I need to find someone to house-sit”
dependent clause: “Although I want to take the dogs on a hike … ”

Like a compound sentence, the sentence contains two unique, independent clauses. It also contains a dependent clause like a complex sentence. Thus it is a compound-complex sentence.
“She took pictures of the children, but their parents, who were still very superstitious, wanted her to delete them.”
independent clause: “She took pictures of the children”
independent clause: (but) their parents wanted her to delete them.
dependent clause: who were still very superstitious

“Summer romances begin for all kinds of reasons, but when all is said and done, they have one thing in common.” – Allie, a character in the movie The Notebook

The woman who visited my sister teaches computer skills.
The book that Jonathan read is on the shelf.
The suburb where I grew up is on the north side of the town.

Compound-complex sentences such as these are very common in English, but one mistake that students often make is that they try to write them without having mastered the simple sentences, compound sentences, and complex sentences first.