There are a LOT of songs with counting in them, from Len Berry’s 1-2-3 and Feist’s 1234 to the Jackson Five’s ABC (“easy as 1, 2, 3”) and the Beatles’All Together Now or the end of You Never Give Me Your Money (“1, 2, 3, 4, 5, 6, 7, All good children go to heaven”). Here’s a Reddit post on the topic, and there are plenty more.
By odd coincidence, I played a couple songs in the category on the same day recently.
13 Question Method by Ry Cooder is a fairly obscure Chuck Berry song. The YouTube description says it was a Berry bootleg in 1957, then released as a legitimate track in 1961.
Count the Days (1-2-3-4-5-6-7), recorded by Gene Pitney was written by Y. Williams, C. Fox, and B. O’Dell. Pitney’s take released at the end of 1968 does not appear to have charted. However, a version from about a year earlier by the brother-and-sister group Charlie & Inez Foxx went to #17 R&B and #76 pop on the Billboard charts. The duo’s big hit was Mockingbird, later covered by James Taylor and Carly Simon.
I must admit being a sucker for a Beach Boys song in the genre, When I Grow Up (To Be A Man), that starts with age 14 and fades out at 31. It got to #9 in 1964.