What is an example of a quasicategory with an outer 4-horn which has no filler? A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy counterexamples.
For n=2, let $X=\Delta[2]$ (which is the nerve of a category): there is a map $\Lambda^0[2]\rightarrow\Delta[2]$ that maps vertices $0\mapsto 1$, $1\mapsto 0$, and $2\mapsto 2$, which can't have a filler because there is no arrow from $1$ to $0$ in $\Delta[2]$.
For $n=3$, any small category $X$ which has a morphism $f$ which is not an epimorphism will give us an unfillable $\Lambda^0[3]$. For example, let $X=N(\Delta)$, and consider the maps $f=d^0\colon [0] \rightarrow [1]$ and $g=d^0\circ s^0\colon [1]\rightarrow [1]$. Then we can construct the horn $\Lambda^0[3]$ where the $d_1$-face witnesses the composition $g\circ f = d^0 \circ s^0 \circ d^0 = d^0 = f$, and the $d_2$ and $d_3$ faces witness $\text{id}_{[1]}\circ f = f$. If there were a filler for this horn, then we'd have $\text{id}_{[1]}=g\circ\text{id}_{[1]}=g$, but $g=d^0\circ s^0$ is not the identity.
For $n\geq 4$, a counterexample can't come from the nerve of a category, because nerves of categories are $2$-coskeletal. I've tried to look at other examples of quasicategories—for example, those given in section 8 of Rezk's notes on quasicategories—but I believe I can show the existence of outer horns for $n\geq 4$ in every example given there.
So, are there any known examples of quasicategories with an unfillable $\Lambda^0[4]$?
 A: Here's a way to cook up lots of examples. Notice that $\Lambda^n_0 = \Delta^0 \star \partial \Delta^{n-1}$ so that a functor $\Lambda^n_0 \to \mathcal{C}$ is the same as a functor $\partial\Delta^{n-1} \to \mathcal{C}_{x/}$ where $x$ is the image of $\Delta^0$. If we require that the image of the vertices under this functor all map to the same object $y$, then this factors through the fiber over $y$ which is the Kan complex $\mathrm{Map}_{\mathcal{C}}(x,y)$. 
So a map $\Lambda^n_0 \to \mathcal{C}$ with this behavior on vertices will have a filler if and only if the resulting homotopy class $S^{n-2} \to \mathrm{Map}_{\mathcal{C}}(x,y)$ is null.
Well now it's easy to cook up examples! Take $\mathcal{C} = \mathsf{Spaces}$, $x = *$, $y = S^{n-2}$, and the functor corresponding to some equivalence $\partial\Delta^{n-1} \simeq \mathrm{Map}_{\mathsf{Spaces}}(*, S^{n-2})$.
A: Let $A$ be any $(2,1)$-category containing a composable pair of morphisms $f\colon a\to b$ and $g \colon b \to c$, and a $2$-cell $\alpha \colon g \to g$ such that $\alpha f = \mathrm{id}_{gf}$. Then the Roberts--Street--Duskin nerve $NA$ of $A$ contains a $(4,0)$-horn, described below, which admits a filler if and only if the $2$-cell $\alpha$ is an identity.
To describe this horn, let's first label a few simplices in $NA$. Let $\beta$ and $\gamma$ denote the $2$-simplices in $NA$ given by the $2$-cells $\alpha \colon g \to 1_c\circ g$ and $\mathrm{id}_{gf} \colon gf \to g\circ f$ in $A$. Let $\varepsilon$ denote the $3$-simplex in $NA$ whose boundary faces are
$d_0(\varepsilon) = \beta$, $d_1(\varepsilon) = s_1(gf)$, $d_2(\varepsilon) = d_3(\varepsilon) = \gamma$; this $3$-boundary extends to a $3$-simplex in $NA$ thanks to the equation $\alpha f = \mathrm{id}_{gf}$ in $A$.
The $(4,0)$-horn of interest is as follows:
$$(-,s_1(s_1(gf)),\varepsilon,s_2(\gamma),s_2(\gamma)): \Lambda^0[4] \to NA.$$ By $3$-coskeletality of $NA$, this horn extends to a $4$-simplex in $NA$ if and only if the $3$-boundary $$(s_0(s_0(c)),\beta,s_1(g),s_1(g)) : \partial\Delta[3] \to NA$$ extends to a $3$-simplex in $NA$, which is the case if and only if the $2$-cell $\alpha$ in $A$ is an identity.
