I have an old iMac (early 2006 17-inch) that I’d like to set up as a fancy typewriter for the kids. However, the optical drive is busted, so I can’t use that.
Is there a way to install the latest OS it will run (I think it’s 10.6)?
I’ve tried sharing an external DVDdrive from my 2017 iMac, but for some reason it won’t connect.
Intel iMacs started in 2006. If you have one of those it has a pretty slow Core 2 Duo processor in it, and possible less than 1Gb RAM.
If you have one of those first Intel iMacs, and you have 1Gb RAM installed, you can use 10.6.8 Snow Leopard; with less than 1Gb RAM it has to be 10.5.8 Leopard.
It has 1.5 Gb RAM, so Snow Leopard seems possible. The question is how to get it on there when the disk drive is busted.
I’ll go on a google-quest 
Perhaps I could create a disc image on my working iMac and then mount that on the old iMac…
Snow Leopard on disc, for around $15 on eBay.
Apple still sells the SuperDrive, for $79. Otherwise you can get it on eBay for less than half that. But OWC/MacSales has 3rd-party new external optical drives starting at $35:
Years ago I just purchased an external USB DVD drive on amazon for around $25. I think it’s an LG.