As machead and mayer say, every Mac OS/iOS device has a software floor - it won't run an operating system that's older than the hardware itself.
But let's step back for a moment, and look at the trigger for your desire: What are you doing with your current computers such that you think a new computer with a new OS won't be "perfectly compatible"? What computers are you using now, and what operating system are they running? What does "perfectly compatible" mean in your situation?
A new computer with a new OS will connect to your network, share files with your existing computers, and use the networked printers you already own. Are there specific applications that you're worried about? If you have a particular application that won't run under OSX 10.10, you can run an older operating system (OSX, Windows, Linux, Solaris) as an application on top of OSX by using emulator software such as Parallels, VMWare Fusion or VirtualBox. The primary intention of these products is to run Windows applications on Mac (or older versions of Windows on newer versions), but you can also run versions of OSX as old as 10.6 on Macintosh hardware that otherwise wouldn't support them. Running OSX 10.6 requires 10.6 Server (available for $19.99 only by phoning Apple's sales number at 1-800-MY-APPLE); running OSX 10.7-10.9 can be done by using the setup wizards in the virtualization software.
Once you've installed a virtualized OS, you then install the applications you need inside the virtualized environment. 10.6 Server/.7/.8/.9 behave as if the main operating system doesn't exist.
This isn't how you'd want to work all day, but for older applications that you don't use all the time it can be a very effective technique. There's a slight speed hit from the extra layer of software; but since it's running on faster hardware to begin with, it's usually a wash.