I downloaded a game called Brain Exercise with Dr.Kawashima. It contains only one executable and no data files. However, it's still possible to save the game!
Where does the save data go?
It may be making saves in the Registry or somewhere under your %APPDATA% folder, or in your homedir's "My Documents" or "My Games" subfolders. Or anywhere else on your hard drive, for that matter. It is unlikely to be modifying its own exe to save, although I suppose that is possible (if a very bad idea: see AttackingHobo's comment below for a sample of the reasons why).
When looking into this kind of thing on Windows, I tend to use tools like Sysinternals Process Monitor (aka
procmon.exe), which will display filesystem and registry activity (among other things, it is verbose -- spend some time filtering down to what you want to see).
While it's perfectly possible to modify the executable (it's what viruses do all the time, and you can append whatever you want to an exe and read from the end of the file - not sure how windows user levels interfere with it, haven't done it since the golden days of MS-DOS) I'd indeed suspect the registry is used.
Copy the executable over an install on another system and see if it has your savegame, if it does you know it's the exe being modified.
While it might not have been that common when this question was originally created, some games today store their savegames
on a server connected to the internet in The Cloud™. This usually requires that the user authenticates itself with a username and password. When the game itself has no account management, a digital distribution system can provide this. Steam, for example, offers an API for storing game data on their servers.