This question depends heavily on individual resources: whether one is willing to buy, store, and, perhaps most importantly, maintain multiple bikes. (But to answer the question briefly, the best minimum is "one of each".)
In the long term the most important point is likely a recent observation made on bicycles.SE: the maintenance required is proportional to the number of kilometers ridden, not to the number of bikes.
The fact that there isn't one answer that would fit everyone doesn't mean that there aren't enough options to satisfy each cyclist, perhaps with the added benefit of finally appreciating what the (n+1) is about.
There are many options
Without a current road bike
If you don't already have a road bike, the following options would be workable.
The true-minimum option
Get a gravel bike with 700c-38 tires. Buy a set of studded 700c-38 tires. Swap tires in November and in March.
That would not be a terrible solution, but it's hardly a great one:
- 38mm tires would be too narrow for the third picture above.
- It takes >60% more power to maintain the same speed on gravel tires at their nominal 40psi as it does to ride on road tires inflated to 90psi.
(The option of using just a fatbike with its knobby tires in the summer and with studded tires in winter would be a miserable solution. It means one would have to put up with clicking sounds year-round: from knobby tires in the summer and from studs in the winter.)
The convenient-minimum option
Same as the previous solution, but buy a new set of rims + a duplicate cassette. Swap tires already mounted on rims in November and in March. (That's basically what motorists do, installing winter tires with a second set of rims.)
The impossible option
Get a gravel bike (or any bike with no suspension) that can accommodate 29"x2.25" on 700c tires. Such bikes are just barely starting to exist.
Use the narrowest tires that'll fit on the rims in the summer, and 29"x2.25" in winter, on one set of rims + cassettes—or the skinniest tires you fancy on a different set of rims + cassettes.
That may be just enough to go on deep snow.
With a current road bike
If you are already happy with your road bike, a gravel bike may not contribute enough to start cycling on snow and ice.
The approximate solution
Get studded 2.25"-wide tires.
Install them on a hardtail, preferably with 29" wheels. The front suspension is pointless, but a hardtail is the only frame that will fit (29")x2.25" tires.
Use the road bike, unless there is snow or ice. When either is on the ground, use the mountain bike with studded tires.
The near-best option
Fit studded 29"x2.25" tires on a mountain bike. Get a fatbike with 26"x4.0" tires.
Ride road during the summer and for the first picture, mountain with studded tires for the second, and fatbike with regular fatbike tires for the third.
The justification here is that after heavy snow fall the fat tires will float just fine, and there will be less risk of hitting an ice patch (and so it may be possible to avoid also getting studded fatbike tires).
If the temperature has risen enough for snow to have melted, then fallen back quickly, there may be patches of ice on the road. If it then snows, the ice will be invisible, but will be just as slippery. If you're too concerned about this scenario, the following option is unavoidable.
The best option
Again, fit studded 29"x2.25" tires on a mountain bike, but also fit studded 26"x4.0" tires on a fat bike.
Ride road during the summer and for the first picture, mountain with studded for the second, and fatbike with studded for the third.