The last version of the OS to use this font as default was System 7.
For Mac OS 8, Apple decided to discard this wonderful part of their
brand’s identity, and commissioned David Berlow of the Font Bureau to
Apple actually released System 7 for free download here. Conceivably you could load up System 7 in an emulator like Sheepshaver and get the font out somehow. Or unpack the system images provided by Apple, though I don't know which software you would need for that.
Apple abandoned Chicago as of OSX, as evidenced by the outrage here. Locally, I can buy an original iMac running OS 9 for 50 EUR. So you could buy one of those and transfer the font :-)
UPDATE: extracting Chicago from System 7
- Download the System 7 disk image from Apple for free here
- Unpack System 7.0.1.smi.bin with "The Unarchiver.app" (free on the App Store here)
- Convert System 7.0.1.smi to a dmg image using Disk Utility using the menu item 'Images > Convert'
- Open the resulting System 7.0.1.smi.dmg and get the Fonts.image inside
- Convert Fonts.image to a dmg using Disk Utility
- Open the resulting Fonts.image.dmg
- Enjoy your retro goodness.