Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
(NB: PowerPC applications can still be run in Snow Leopard using Rosetta, but they will not work in later versions of OS X. For detailed information see this user tip: https://discussions.apple.com/docs/DOC-6271 )
The OS 10.6 Snow Leopard install DVD is still available for $19.99 from the Apple Store:
and in the UK:
but like Leopard, Snow Leopard is no longer supported by Apple in terms of security updates.
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard to bring it up to 10.6.8, or download the combo update from here:
Check via Software Update whether any further updates are required, particularly to iTunes (you will need the latest version in order to synch with mobile devices).
You should now see the App Store icon in iTunes, and you now need to set up your account:
You can now upgrade to Mavericks OS 10.9 for free IF you have one of the following Macs, with not less than 2GB of RAM, and at least 8GB of available space on your hard drive:
iMac (Mid-2007 or later)
MacBook (13-inch Aluminum, Late 2008), (13-inch, Early 2009 or later)
MacBook Pro (13-inch, Mid-2009 or later), (15-inch, Mid/Late 2007 or later), (17-inch, Late 2007 or later)
MacBook Air (Late 2008 or later)
Mac Mini (Early 2009 or later)
Mac Pro (Early 2008 or later)
Xserve (Early 2009)
iCloud system requirements:
If you cannot run Mavericks you can purchase the code to use to download Lion from the App Store (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard):
or Mountain Lion:
I think you may have started us on a dead-end path with your equipment description. Your line says "Core Duo" which cannot run any higher OS than 10.6 Snow Leopard, but there are no Core Duo iMacss running at 2.66ghz, just Core 2 Duos that can run at least OS 10.7.
Also the first 21-inch iMacs had either 3.06 or 3.33Ghz Core 2 Duos.
So we need to clarify some conflicting information. Please use this Apple support page to enter your computer's serial number:
A link near the top of the text section leads to a place to input a serial number.
If your comptuer is too old to work with that lookup function, you can try a third-party site where you can enter model and order numbers you can find on your computer's base or foot:
At this point we are guessing about your iMac variant. With the extra information, we can give you a definitive answer.
Allan's page says to plug the serial number into Apple's own serial number lookup.
The page he references says:
By serial number
On most computers with Mac OS X, you can find the serial number in the About This Mac window. Otherwise, seeiMac: How to locate the serial number to learn where to find the serial number on the surface of your iMac.
Type the serial number into the Check Your Service and Support Coverage page. The resulting screen will show your model.
When you go to the the Check your Service and Support Coverage, that's a good link to find that info.
I think you mean 4 GB of RAM. 4 MB of RAM was not standard since the early 1990s!
Yes, you have enough computing capacity for 10.9.2
https://discussions.apple.com/docs/DOC-6271 will tell you if your software is up to the ordeal.
Not sure if we should encourage the publication of serial numbers. Do you know something we may not be aware of?
I agree. That's why I like the Apple page. In cases where the Apple lookup doesn't return any findings, I post the everymac.com page because it has the option of using a model, order, or EMC number instead of a serial number.