What does a nonsymmetric connection on $\Bbb R^n$ look like?

One of the simplest examples I can suggest is this (although it's not on Euclidean space). A compact Lie group (e.g., $SO(3)$) has a unique (up to scaling) bi-invariant metric. For the associated (symmetric) Riemannian connection the geodesics through the identity element are one-parameter subgroups, and in general they're just group translates of one-parameter subgroups. The connection is given by $\nabla_XY = -\frac12[X,Y]$ for left-invariant vector fields $X,Y$, and the curvature tensor can be computed algebraically from the Lie bracket.

However, we can easily define an invariant flat connection by setting the covariant derivative $\nabla_XY=0$ for any left-invariant vector fields $X,Y$. Then the torsion tensor $\tau$ will be given by the Lie bracket, i.e., $\tau(X,Y) = -[X,Y]$. So now all the Lie algebra is in the torsion.

You probably can't detect the difference with geodesics (since any invariant vector field is parallel along its integral curves either way), but you sure can see the difference with general parallel translation.

EDIT: Here's a connection $\nabla$ on $\Bbb R^2$ so that the parabolas $y=x^2+c$ are geodesics. Let $X_1 = \dfrac1{\sqrt{1+4x^2}}(1,2x)$ and $X_2 = (0,1)$, and set $\nabla X_1 = 0$, $\nabla X_2 = \left(\dfrac 2{\sqrt{1+4x^2}} X_1 - \dfrac{4x}{1+4x^2} X_2\right)\otimes dx$. The connection is metric-compatible and definitely not symmetric.

If we write $g_{ij} = X_i\cdot X_j$ (using the usual $\Bbb R^2$ dot product) and $\nabla X_i = \sum_j \omega_i^j\otimes X_j$, metric compatibility is given by the equations $$dg_{ij} = \sum_k g_{ik}\omega_j^k + g_{kj}\omega_i^k.$$ Here we have $\omega_1^1 = \omega_1^2 = 0$, $\omega_2^1 = \dfrac2{\sqrt{1+4x^2}}dx$, and $\omega_2^2 = -\dfrac{4x}{1+4x^2}dx$, and you can check that compatibility follows. Note that the connection is obviously not symmetric, as \begin{align*} \nabla_{X_2}X_1 &= 0, \quad \text{and yet} \\ \nabla_{X_1}X_2 &= \omega_2^1(X_1)X_1 + \omega_2^2(X_1)X_2 = \frac2{1+4x^2}X_1-\frac{4x}{(1+4x^2)^{3/2}}X_2. \end{align*} Last, since $\nabla_{X_1}{X_1} = 0$, the integral curves of $X_1$ are geodesics. These are the parabolas $y=x^2+c$.

I asked the question. However, I think I have a simpler example than the currently accepted answer.

Let $X=(1,0)$ and $Y=(0,1)$ be the constant unit vector fields in the $x$ and $y$ directions. Define a connection such that: $$\begin{matrix}\nabla_XX=Y&\nabla_XY=-X\\ \nabla_YX=0&\nabla_YY=0\end{matrix}$$ Equivalently, $\Gamma_{12}^1=-1$, $\Gamma_{11}^2=1$, and all other Christoffel symbols are zero. Clearly, it is not symmetric.

Parallel transport of a vector along a curve is equivalent to rotating the vector by an amount equal to the horizontal displacement of the curve. Since rotation is an isometry of tangent spaces, this property tells us that the connection is compatible with the metric.

Since $\nabla_YY=0$, vertical lines are geodesics. The other geodesics each look like a component of the graph of $\ln(\cos(x))$, but translated some amount horizontally and vertically. (This can be derived from the aforementioned property of parallel transport.) In other words, the nonvertical geodesics look like components of the graph of $\ln(\cos(x+C_0))+C_1$ for some $C_0,C_1$.

To check this, we can take $C_0=C_1=0$ without loss of generality. We need a unit-speed parametrization of $y=\ln(\cos(x))$. It can be checked that the following is such a parametrization: \begin{align}x(t)&=\tan^{-1}(\sinh(t))\\ y(t)&=-\ln(\cosh(t))\end{align}

The equations for a geodesic, in this case, demand: \begin{align}\frac{{\rm d}^2x}{{\rm d}t^2}-\frac{{\rm d}x}{{\rm d}t}\frac{{\rm d}y}{{\rm d}t}&=0\\ \frac{{\rm d}^2y}{{\rm d}t^2}+\left(\frac{{\rm d}x}{{\rm d}t}\right)\!^2&=0\end{align} Calculation shows that $\frac{{\rm d}x}{{\rm d}t}=\operatorname{sech}(t)$ and $\frac{{\rm d}y}{{\rm d}t}=-\tanh(t)$, and the result follows.

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EDIT: There's another interesting property of this connection. If we define $\nabla'$ such that ${\Gamma'}_{ij}^k=\frac12(\Gamma_{ij}+\Gamma_{ji})$, then we end up with a symmetric connection with the same geodesics. However, parallel transport changes, and it is no longer compatible with the metric. In fact, $\nabla'$ is not compatible with any metric!

To see this, note that there is no geodesic that goes between $(0,0)$ and $(\pi,0)$, or between any pair of points whose horizontal distance is more than or equal to $\pi$. If $\nabla'$, which is symmetric, were compatible with any metric, then it would be the Levi-Civita connection on the corresponding Riemannian manifold. This would contradict the Hopf–Rinow theorem, which states that, for any complete Riemannian manifold, there exists a geodesic between any two points.

(This also means that the Hopf–Rinow theorem crucially uses the symmetry of the connection, since it fails with our asymmetric $\nabla$ as well.)