One would have to be a looney to choose cartridges based on "no common chamberings" criteria.
Using firearms in the safe, though, I can come close:
.22 Short,
Remington Model 24 (mfg 1931) and High Standard Model C (mfg 1940), both bought NIB by my paternal grandfather.
6.5-06AI,
Interarms Mark X action, Boyd stock, Timney trigger, action trued with Krieger 24" fluted stainless barrel with "Coyote Hunter" engraved on right side.
.45-70,
Marlin 1895, standard length barrel, no ports. 460g Cast Performance WFNGC @ 1812fps, 3355fpe at the muzzle for DG, 300-350g for non-DG
Being more pragmatic and again using firearms in the safe, I'd choose these:
.22LR,
Browning SA-22, Browning Buckmark
7mm Remington Magnum,
Ruger M77/RS (factory iron sights) and 3-9x scope
.338WM,
Ruger MKII/RS (factory iron sights) and 3-9x scope, semi-custom assembled from stainless action, stainless barrel shortened to 22" and fluted by previous owner, Ruger skeleton boat paddle stock
