American, British, and French forces will exercise at the corps level.
Warfighter 21-4 is interesting:
Warfighter 21-4, a corps-level combat exercise starting April 6, will involve U.S., French and British troops, a U.S. Army press release said this week. ...
The emphasis will be on interoperability, as British and French forces working within a U.S. corps.
The combined formations will be integrated at the brigade-level with a British brigade operating within a U.S. division, and U.S. brigade combat teams in British and French divisions.
Britain and France are probably the only NATO countries that could provide divisions in war that American brigades could in theory operate under. The Turks could too, but the politics these days rule that out even in theory.
But America is the only country that could really field corps that could include allied units within them.