Which high-degree derivatives play an essential role? 
Q. Which high-degree derivatives play an essential role
  in applications, or in theorems?

Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the 3rd derivative (jolt or jerk) certainly play several roles in applications.
And the torsion of a curve in $\mathbb{R}^3$ can be expressed
using 3rd derivatives.
Beyond this, I'm out of my depth of experience. I know of the biharmonic equation
$\nabla ^4 \phi=0$.
There is a literature on the solvability of quintics,
but it seems this work is neither aimed at applications nor essential to 
further theoretic developments. 
(I am happy to have my ignorance corrected here.)

Q. What are examples of applications that depend on 4th-derivatives 
  (snap/jounce) or higher?
  Are there substantive theorems that require existence of $\partial^4$ or higher as
  assumptions, but do not require (or are not known to require) smoothness—derivatives of all orders?

 A: 
What are examples of applications that depend on 4th-derivatives ...
  or higher?

The Dirac equation is a system of four partial differential equations for four complex functions. However, in a general case, it is equivalent to one fourth-order equation for just one function (see references to my article).  
A: In geometric measure theory (GMT), many theorems require more than four derivatives. This is in part due to the use of Nash embedding theorem (which require $C^3$). We will give two examples.
Almgren's big regularity theorem (1983, 2000) required the manifold to be $C^5$, but De Lellis and Spadaro (2014) managed to reduce the needed regularity to $C^{3,\alpha}$ for some $\alpha>0$ (they also simplified and shortened a lot the proof). 
In Almgren-Pitts theory (1976-present day), the original result of Pitts (1981) concerning the existence of one closed minimal hypersurface in closed Riemannian manifolds of dimension $3\leq n\leq 6$ required the manifold to be $C^k$ with $k\geq \mathrm{max}\{4,n-1\}$ (and the minimal hypersurface will also be of class $C^k$).
A: Moser’s theorem in (1962) famously required estimates on the first 333 derivatives.
A: A robotics application (similar to the one given by Martin Fevre) involves minimizing a function of the snap of a quadrotor's trajectory, where $\mathrm{snap}(t) :=\frac{\mathrm{d}^4x}{\mathrm{d}t^4}$; here is a reference.
A: This is probably a stretch, but the theory of distributions is highly relevant to a large chunk of applied math and heavily relies on functions which have derivatives of arbitrarily high order.
A: There is a famous story that Richard Nixon once made use of the third time derivative to support his re-election, via a claim that the rate of increase of inflation was decreasing.
http://www.ams.org/notices/199610/page2.pdf
A: I was once president of a PhD defense in chemistry (this is standard in our faculty; the president should not belong to the same department as the student). Most of his thesis involved simulations of behaviours of the seventh derivatives of something. It was claimed that this was testing for chaotic chemical reactions (certainly periodic reactions exist). The student however, was mathematically quite weak, so I don't really know how effective his results were. 
A: I think third-order and fourth-order derivatives in particular are very common in both pure and applied mathematics.  For example, I once worked with some equations modelling the local film thickness of a moving liquid film which contained third-order derivatives:
$\delta q_{t} = \frac{5}{6}h - \frac{5}{2}\frac{q}{h^{2}} + \delta \Bigg( \frac{9}{7}\frac{q^{2}}{h^{2}}h_{x} - \frac{17}{7}\frac{q}{h}q_{x} \Bigg) +  
  \frac{5}{6}hh_{xxx} + \eta \Bigg[ 4\frac{q}{h^{2}}(h_{x})^{2} - \frac{9}{2h}q_{x}h_{x} - 6 \frac{q}{h}h_{xx} + \frac{9}{2}q_{xx} \Bigg].$
A: The error in Simpson's rule
for integration is usually expressed in terms of the fourth derivative of the integrand.
A: In
"classical 
(Euler-Bernoulli) beam theory" the motion of a beam
is modelled by the 4th-order PDE 
$$
EI \frac{\partial^4 w}{\partial x^4} 
= -\mu \frac{\partial^2 w}{\partial t^2} + q.
$$
A: Nonlinear solitonic wave equations often feature high($3^+$) order of derivatives.
The most famous one may be the KdV equation: $\partial_t \phi + \partial_{xxx} \phi -6 \phi \partial_x \phi = 0$.
There is also the Boussinesq equation $\partial_{tt} \phi - \partial_{xx} \phi -\alpha \partial_{tt} (\phi^2) - \partial_{xxxx} \phi = 0$.
A: I agree with Dirk's answer that there exist also applications for arbitrarily high order derivatives. Another example is the infinite order Kosterlitz–Thouless phase transition
A: In optimal transport, there is a quantity known as the MTW tensor [1] which depends on the fourth derivatives of the cost function. The regularity theory of transport depends in a crucial way on the sign of this tensor. Unless the MTW tensor non-negative definite, the optimal map can fail to be continuous, even when the measures are smooth and satisfy a necessary convexity condition [2]. As a result of this,  much of the regularity theory depends on the cost function being $C^4$, although it is possible to make some of the theory work when the cost function is $C^3$ [3].
For a more complete story, there is a survey paper of De Phillipis and Figalli which gives a good overview of the theory [4]. 
[1] Ma, Xi-Nan; Trudinger, Neil S.; Wang, Xu-Jia, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177, No. 2, 151-183 (2005). ZBL1072.49035.
[2] Loeper, Grégoire, On the regularity of solutions of optimal transportation problems, Acta Math. 202, No. 2, 241-283 (2009). ZBL1219.49038.
[3] Guillen, Nestor; Kitagawa, Jun, On the local geometry of maps with c-convex potentials, Calc. Var. Partial Differ. Equ. 52, No. 1-2, 345-387 (2015). ZBL1309.35038..
[4] De Philippis, Guido; Figalli, Alessio, The Monge-Ampère equation and its link to optimal transportation,  ZBL06377770. 
A: Analytic regularity of a $C^\infty$ function can be characterized by using an  infinite number of derivatives. A function $f\in C^\infty(\Omega)$ where $\Omega$ is an open subset of $\mathbb R^n$ is real-analytic on $\Omega$ iff for all $K$ compact subsets of $\Omega$, there exist positive constants $C_K, \rho_K$, such that 
$$
\begin{array}{ccc}
\forall \alpha=(\alpha_1, \dots, \alpha_n) \in \mathbb N^n \,,
& ~~ &
\sup\limits_{x\in K}\left|(\partial_x^\alpha f)(x)\right| \le C_K \rho_K^{-\vert \alpha\vert} \alpha ! \,,
\\[10px]
\vert \alpha\vert=\sum \alpha_j \,,
& &
\alpha!=\prod \alpha_j! \,.
\end{array}
$$
A: The Kuramoto-Sivashinsky equation
$$\partial_tu+\Delta^2u+\Delta u+\frac12|\nabla u|^2=0$$
where $\Delta$ is the Laplace operator (second order) was derived to model diffusive instabilities in a laminar flame. There is a huge literature about it.
A: Third derivatives (and higher) show up naturally in the study of affine and projective geometry.  Perhaps most notably in the Schwarzian.
The Bochner formula also involves three derivatives and is a fundamental tool in geometric analysis.  There are a number of important computations in the same spirit that sometimes require four derivatives (e.g. Simons' identity for minimal surfaces).
A: As a roboticist, I always pay close attention to the 3rd derivative of position with respect to time when generating a motion for a given robot. The 3rd derivative of position is most often referred to as the "jerk" and quantifies the smoothness of a motion. See http://courses.shadmehrlab.org/Shortcourse/minimumjerk.pdf article for a great description of jerk. 
In general, people will try to minimize the jerk to obtain a smooth motion for the robot. If position as a function of time is specified by $x(t)$, then the jerk is
$\dddot{x}(t) = \frac{d^3 x(t)}{dt^3}$
A: In Kahler geometry, and many other places, (Riemannian) metrics are often (sometimes) prescribed via a potential function, which involves two derivatives.  Therefore, the fundamental invariants, like the Riemann curvature tensor, involve fourth derivatives of this function.  
As for arbitrarily high derivatives, one place these arise naturally is the study of Lie groups $G$ and their homogeneous spaces $G/H.$  For instance, one may ask if there is a "differential operator" acting on locally defined diffeomorphisms of $G/H$ whose kernel consists exactly of those arising from the left $G$-action.  As noted by RBega2 above, for $G=PSL(2, \mathbb{C})$ and $G/H=\mathbb{CP}^{1}$ one is led to the Schwarzian derivative, a non-linear third order differential operator. 
This question was studied in general by Spencer in the 1960's resulting in the so-called Spencer complex.  
Nonetheless, both of these cases might be more high brow than the OP was looking for.
A: The OP says

And the torsion of a curve in $\mathbb R^3$
  can be expressed using 3rd derivatives.

More generally, 

a curve in $\mathbb R^3$ is described up to isometry by the derivatives up to order $3$.

But then is it not also true that

a curve in $\mathbb R^4$ is described up to isometry by the derivatives up to order $4$.

and

a curve in $\mathbb R^5$ is described up to isometry by the derivatives up to order $5$.

and so on?
A: Similarly to Martin Fevre's answer: 
When routing a road for fast vehicles you need to minimize the rate of change of the road curvature so that the drivers wouldn't have to move the steering wheels in quick jerks. For example, a perfect transition from a circular roundabout to a linear road tangent to the roundabout would require an instantaneous steering wheel adjustment; that can be avoided by connecting the circle and the line with a ramp that has smoothly varying curvature. 
Since the curvature is essentially the 2nd derivative the rate of change of the curvature is the 3rd derivative, minimizing the rate of change of the curvature requires the 4th derivative.
A: Given two sets $A$ and $B$ in $\mathbb{R}^n$, the Minkowski sum written $A+B$ is the set $\{a+b:a\in A,b\in B\}$.
If $A$ and $B$ are convex subsets of $\mathbb{R}^2$ with real-analytic boundaries then the boundary of $A+B$ is only guaranteed to be '$6\frac{2}{3}$ times differentiable,' by which I mean $6$ times differentiable with $6$th derivative Hölder continuous with exponent $\frac{2}{3}$. This is known to be sharp.
A: To perhaps state the obvious, there are many places in theory (and, I'm told, practice) where analytic continuations are used.  For instance, the Riemann zeta function was discovered as the analytic continuation of Euler's product formula.  
Of course you can't make any sense of this without all the derivatives of a function.
