Chromatic homotopy tends to mainly focus on $1$-dimensional (commutative) FGLs. From a geometric perspective, this is because line bundles form a group and n-plane bundles don't, so the first Chern class gives us a FGL and higher Chern classes do not. At any rate, this appears all over CHT. The Morava E-theories come from Artin-Mazur FGLs, which are $1$d (at least for Calabi-Yau varieties); the Landweber exact functor theorem only applies to $1$d FGLs; and the queen of all complex-oriented cohomology theories, $MU$, is considered to be a spectral version of the universal $1$d commutative FGL.
The closest thing to increasing dimension that I can find is the fact that the $r$th Artin-Mazur FGL $\Phi^r$, which gives rise to the $(r+1)$th Morava E-theory $E(r+1)$, is naturally considered as a formal $r$-group rather than a formal $1$-group. (Technically, it's the $r$-group of line $r$-bundles on a variety of dimension $r$.) But this is more of a progressively higher categorification than anything involving actual higher-dimensional group schemes. Indeed, in the $r\to\infty$ limit (if I'm not mistaken) it just goes to the spectral stack $\operatorname{Spet} MU$, which doesn't recover anything new dimension-wise: it's just the universal one-dimensional spectral formal group law.
Why am I asking about this, you may wonder? The reason comes from algebraic geometry and no small part mathematical physics. In dimensions $1$ and $2$ (and presumably in some fashion in higher dimensions), the Torelli theorem says that sufficiently nice varieties can be recovered from their principally polarized Jacobian/Picard variety. For genus $1$ curves, this principal polarization corresponds to choosing the basepoint of an elliptic curve, an operation which is crucial in relating "QFT over a complex torus" to "the cohomology theory associated to the formal completion of a complex torus". (Skipping over the details with torsors and whatnot, suffice to say that tori don't come with a basepoint, and you have to give them one to make sense of this stuff.) I believe that Stolz and Teichner's program implies that something similar should be true for QFT on K3 surfaces and K3 cohomology, and indeed for higher-dimensional Calabi-Yau varieties and corresponding Morava E-theories. But to generalize beyond this case (e.g. to consider worldsheets other than "a pair of strings spontaneously appear then annihilate") requires the consideration of higher-dimensional groups, since more complicated cohomology = more complicated Picard variety.
