Bluey is a godsend in our household. It's not annoying and the amount of classical and original music woven into each episode is wonderful. There's one particular episode - "Sleepytime" - which I cannot bring myself to watch anymore because I get so emotional. Every parent I talk to with young kids who has seen the episode agrees and knows what I mean. It's really that good, and despite being a fun little show there…
Not just classical. There is an episode which has Futurama's "Pop a poppler in your mouth..." as a background music.
But it could just be that the Poppler song is also based on some classical tune. ;)