i did not understand that question this way at all - what You say is most easily asked for - what game required more than 1.38MB data accessible at the same time to run; even the scenario You describe does not really really require a hard disk;
may OPs question should be understood as the first game with an installer?
Let me clarify my question as I intended it.
First of all, for the sake of this discussion, let's disregard the existence of the 2.88 MB Extra Density diskette format, since both disks and drives of that format were quite uncommon and mostly limited to certain IBM PS/2 models, if I'm not mistaken. Besides, as far as I know, no games were ever distributed on that format, for that very same reason.
So what I'm looking for is the first game that did not fit on a single 1.44MB diskette and did not support run-time disk swapping either. Also, let's leave out any games that came on two disks and supported reading the first disk from the A: drive and the second disk from the B: drive in dual disk drive systems. (Although that would be an interesting special category of games by itself, because I don't know of any games off the top of my head that actually made use of dual floppy drives.) So yes, as others have pointed out, this would most likely have been a game that came with an install or setup utility that would install it to a hard drive before it could be played. Although I guess it could also at least theoretically have been a game that would require the user to manually copy the contents from more than one floppy disk to a single directory on the hard drive.
With so many games released back in the day, I guess it would be difficult to determine which specific game was the first that at the time could only be played on a system with a hard drive. I guess the original boxes of the games (and perhaps also bundled README files) would probably have explicitly stated a hard drive as being required as opposed to recommended.
By the way, I know of a one theoretical workaround for such a requirement: if a PC that lacked a hard drive in those times had an Expanded Memory (EMS) board with a large enough amount of RAM on it installed, then by loading a Virtual Disk driver backed by EMS and installing the game to the virtual disk, it would still be possible to play the game on such a system. It would load a lot faster too, although it would have to be reinstalled after each reboot. 😉 I remember my Dad's 8086 PC being upgraded with an EMS card before it was upgraded with a hard drive. I remember playing with virtual disks, trying to get games to load faster. In the case of Sierra adventures that used the SCI engine, the EMS card would reduce the amount of floppy swapping required by caching the loaded sceneries in EMS, which those games supported natively. No virtual disks needed for those games. 😀