This post is based on the observation that, in a variety of fields (e.g. logic, computics, mathematics, physics), while certain classes of problems can be parameterized by some natural number $n$, it appears that the interesting problems―not so simple as to be trivial, but not so complex as to be unsolvable, undecidable, intractable, or nonexistent―always occur at small $n$. Below is a collection of such problems, describing at which $n$ they are interesting, and how so.
A Summary
 Firstorder predicate logic
 Rank1, rank2 polymorphic lambda calculus
 System F$_3$
 3SAT, 3D matching, graph 3colouring
 (Semi)regular polytopes in 3D and 4D
 Exotic $\mathbb{R}^4$ and exotic 4spheres
 2body problem
 Stable bound orbits in 2D and 3D
Logic
Higherorder logic can be stratified into $n$thorder logics, beginning with propositional logic $(n = 0)$, firstorder logic, secondorder logic, and so on. As opposed to propositional logic, firstorder logic can simulate computation (either by encoding a Turing machine as a formula or by representing recursive functions in a sufficiently rich arithmetic theory), and is therefore undecidable. However, FOL is both compact and admits a sound and complete finitary proof system, whereas secondorder and higherorder logics do not. Therefore, FOL is interesting by being sufficiently complex while still wellbehaved.
Computics
Polymorphic Lambda Calculus
The polymorphic lambda calculus System F can be stratified into systems of rank$k$ polymorphism, where forall quantifiers are restricted to no more than $k$ arrows deep on the left (e.g. ((∀ a. a → t1) ... → tk)
). Beyond rank1 (prenex or Hindley–Milner) and rank2 polymorphism, type inference and type checking is undecidable [1], which makes these interesting in terms of practicality. Rank0 polymorphism is essentially STLC (i.e. nonpolymorphic), where type inference and type checking are trivially decidable.
The higherorder polymorphic lambda calculus System F$_{\omega}$ can be stratified into levels as well, where System F$_{_1}$ corresponds to STLC, System F$_2$ corresponds to System F above, and System F$_k$ allows type constructors with argument types of kind level $(k  1)$. While there exist (Currystyle) lambda terms that cannot be typed in F$_2$ but can be typed in F$_3$ and above [2, 3], it is conjectured [4, 5] that all terms typeable in F$_3$ and above are also typeable in F$_3$ itself; there is a supposedly incorrect proof of this [6]. This is not to say System F$_3$ and above are uninteresting because they are too complex; in fact, it is interesting precisely because higher orders of polymorphism may collapse to F$_3$. This is similar to NPcomplete problems below, where classes of problems parametrized by $n$ are all NPcomplete for $n \geq 3$.
Computational Complexity
There are many decision problems parametrized by $n$ whose complexity is NPcomplete only for $n=3$ or possibly higher. Here are three classical problems that are usually encountered. I won’t list any more simply because there are far too many of their kind.

nSAT: Given a formula in conjunctive normal form where each clause has $n$ literals, is there an assignment of truth values to its variables such that the formula is satisfied? In 0SAT, the answer is trivially yes; 1SAT and 2SAT are in P, while 3SAT and higher are NPcomplete.

ndimensional matching: Given a natural number $k$, $n$ finite disjoint sets, and a subset $T$ of their Cartesian product, is there a subset $M$ of $T$ such that $M > k$ and the elements of the tuples of $M$ are disjoint? 1D matching is trivial and 2D matching (bipartite matching) is in P, while 3SAT and higher are NPcomplete.

Graph kcolouring: Given a graph, is there a way to assign each node one of $k$ colours such that no two nodes of the same colour share an edge? 1colouring is trivial and 2colouring is in P, while 3colouring is in NP. Interestingly, for 4colouring the answer is always yes.
Mathematics
In mathematics, there appear to be many interesting structures in fourdimensional space. This is a vastly incomplete list; there are, as far as I know, some unique properties of 4D space to do with nspheres as well.
Geometry
In $n$ dimensions, there exist at least the following regular polytopes: the simplices, (with $(n+1)$ faces of $(n1)$dimensional simplices), the quadruplices (with $2n$ faces of $(n1)$dimensional quadruplices), and the orthoplices (with $2^n$ faces of $(n1)$dimensional simplices). In 0D and 1D, these all collpase to the trivial point and line, respectively. In 2D, there are infinitely many regular convex and star polytopes (polygons and stars). For dimensions greater than 4, there are no other regular polytopes. However, in 3D and 4D, there exist other regular polytopes than the three categories above. In 3D, the convex polyhedra are the icosahedron and the dodecahedron, and the star polyhedra are Kepler–Poinsot polyhedra. In 4D, the convex polychora are the 120cell (corresponding to the dodecahedron), the 600cell (corresponding to the icosahedron), and the 24cell (with no 3D counterpart), while the star ones are Schläfli–Hess polychora.
Similarly, in $n$ dimensions, there exist at least the following semiregular polytopes: the $k_{21}$ polytopes, and the regular polytopes as described above. In 0D, 1D, and 2D, the semiregular polytopes are exactly the regular polytopes. For dimensions greater than 4, there are no other semiregular polytopes. However, in 3D and 4D, there exist other regular polytopes than the $k_{21}$ polytopes. In 3D, these are the prisms, the antiprisms, and the 13 Archimedean solids (excluding enantiomorphs). In 4D, these are the rectified 600cell (analogous to the icosidodecahedron) and the snub 24cell.
Topology
Exotic smooth structures on $\mathbb{R}^n$ are smooth topological manifolds that are homeomorphic but not diffeomorphic to $\mathbb{R}^n$. It was shown by Stallings [7] that no such structures exist for $n \neq 4$, while it was shown by Taubes [8] that uncountably many exist for $n = 4$. Similarly, exotic piecewiselinear structures on the $n$sphere are not PLhomeomorphic to the $n$sphere. The generalized PL Poincaré conjecture states that there are no such structures; this holds true for $n \neq 4$, but it is yet unknown whether this holds for $n = 4$ (see [9]).
Physics
There are a few observations that point out the uniqueness of three spatial dimensions (and possibly two as well), usually as arguments for the anthropic principle.
In three dimensions, the gravitation potential is propotional to $\frac{1}{4}$ (see below). The $n$body problem asks for a general closedform solution to the paths of $n$ bodies mutually attracted by the gravitational forces due to this potential. A general solution exists only for 2 bodies; with 3 or more bodies, no general solutions exist, although there are many special periodic cases.
Gravitation and Electromagnetism
Given a potential field described by Poisson’s equation, the fundamental solution in $d$ dimensions is a potential propotional to $\frac{1}{r^{d2}}$ (or to $\ln(r)$ for $d = 2$), where $r$ is the distance from the source. This potential applies to Newtonian gravitation and classical electrodynamics. In only $d = 2$ and $d = 3$, there exist stable bound orbits (circular or elliptical); the concept of an orbit is undefined in fewer dimensions. The original result by Ehrenfest [10] (and reëxpressed by Freeman [11] in English) showed that bound planetary orbits and the Bohr model are not stable for $d > 3$.
This has been generalized by Tangherlini [12] to a Schwarzschild field (due to a nonrotating, uncharged mass), with the result that stable bound orbits still do not exist in $d > 3$. Since the Schwarzschild metric is a special case of other metrics (Kerr, for rotating masses; Reissner–Nördstrom, for charged masses; Kerr–Newman, for rotating, charged masses), it is not expected that these will have stable bound orbits in $d > 3$ either. There probably are papers that deal with them explicitly.
On the other hand, it was shown [13] that bound states of a single electron in a $\frac{1}{r^{d2}}$ potential using the Schrödinger equation do exist for $d > 4$. As relativistic effects are usually small corrections, it would be expected that special relativistic equations (e.g. Dirac, Klein–Goron, Proca) would also yield the same results. But then we would delve into the world of quantum field theory and have to consider the strong force, the weak force, and Lagrangians, which is beyond the scope of this post. As a subaside, Tegmark [14] argues for the impossiblity of more or less than one time dimension.
[1] Kfoury, A. J. and Tiuryn, J. “Type Reconstruction in Finite Rank Fragments of the SecondOrder λCalculus”. (1992). Information and Computation, 98:2, pp. 228–257. https://doi.org/10.1016/08905401(92)90020G.
[2] https://www.cis.upenn.edu/~bcpierce/types/archives/1993/msg00026.html.
[3] Giannini, P. and Della Rocca, S.R. “Characterization of typings in polymorphic type discipline”. (1988). Third Annual Symposium on Logic in Computer Science, pp. 61–70. https://doi.org/10.1109/LICS.1988.5101.
[4] https://www.seas.upenn.edu/~sweirich/types/archive/1991/msg00054.html.
[5] Giannini, P., Honsell, F., and Ronchi Della Rocca, S. (1993). “Type Inference: Some results, Some problems”. Fondamenta Informaticae, 19:1–2, pp. 87–125. https://dl.acm.org/doi/abs/10.5555/175469.175472.
[6] Malecki, S. (1997). “Proofs in system Fω can be done in system Fω1”. International Workshop on Computer Science Logic, pp. 297–315. https://doi.org/10.1007/3540631720_46.
[7] Stallings, J. (1961). “The PiecewiseLinear Structure of Euclidean Space”. Proceedings of the Cambridge Philosophical Society, 58:, pp. 481–488. https://doi.org/10.1017/s0305004100036756.
[8] Taubes, C. H. (1987). “Gauge Theory on Asymptotically Periodic 4Manifolds”. Journal of Differential Geometry, 25:3, pp. 363–430. https://doi.org/10.4310/jdg/1214440981.
[9] Freedman, M., Gompf, R., Morrison, S., Walker, K. (2010). “Man and Machine Thinking about the Smooth 4Dimensional Poincaré Conjecture”. Quantum Topology, 1:2, pp. 171–208. https://doi.org/10.4171/qt/5.
[10] Ehrenfest, Paul. (1920). “Welche Rolle spielt die Dreidimensionalität des Raumes in den Grundgesetzen der Physik?”. Ann. Phys., 366:, pp. 440–446. http://doi.org/10.1002/andp.19203660503.
[11] Freeman, Ira M. (1969). “Why is Space ThreeDimensional?”. American Journal of Physics, 37:12, 1222–1224. https://doi.org/10.1119/1.1975283.
[12] Tangherlini, F. R. (1963). “Schwarzschild field inn dimensions and the dimensionality of space problem”. Nuovo Cim, 27:3, pp. 636–651. https://doi.org/10.1007/BF02784569.
[13] Caruso, F., Martins, J., and Oguri, V. (2013). “On the existence of hydrogen atoms in higher dimensional Euclidean spaces”. Physics Letters A, 377:9, pp. 694–698. http://dx.doi.org/10.1016/j.physleta.2013.01.026.
[14] Tegmark, Max. (1997). “On the dimensionality of spacetime”. Classical and Quantum Gravity, 14:4, pp. L69–L75. http://dx.doi.org/10.1088/02649381/14/4/002.