would be; Tiger, Leopard, Snow Leopard, Lion and Mountain Lion. (Expensive!)
However, apparently you are not sure of what you have so the route is even easier.
Your computer is NOT running Panther, it is running Leopard. From there it is a cheap route to Snow Leopard, Lion and Mountain Lion.
You "might" be able to get there by using the cheaper Snow Leopard then Mountain Lion update software but there is a distinct possibility that you will have a problem with that route.
There is also a possibility that you could go straight to Mountain Lion and skip the others but that might present problems too.
Before you go down that route, make sure that you have a complete backup of your system because if anything goes wrong you will almost certainly have to reformat the drive and reinstall 10.5.8 again.
Problem is that Apple no longer sells Snow Leopard or Lion so your only source will be places like eBay or Amazon.
Check out the prices and see what you think.
P