A differential operator of first order with real coefficients cannot be elliptic (unless $n=1$ which we should and do exclude). But if complex coefficients are allowed, then in dimension $2$ there are elliptic first order operators, such as $\frac{\partial}{\partial x_1}+i\frac{\partial}{\partial x_2}$ which you may recognize as a form of the Cauchy-Riemann operator. This is why we can have nice things in complex analysis.
In dimensions higher than $2$ even complex coefficients do not help, since any $\mathbb R$-linear map from $\mathbb R^n$ to $\mathbb C$ has nontrivial kernel. Suppressing the temptation to dust off quaternions (hm, quaternion-valued elliptic PDE?) we admit that any elliptic operator in dimensions above $2$ must have order at least $2$. Like the Laplacian $\Delta$.
But the order can be higher; for example the bi-Laplacian $\Delta^2$ is elliptic and has order $4$. The powers of $\Delta$ provide easy examples of elliptic operators of any even order (and there are many more). But there are no elliptic operators of odd orders $3,5,7,\dots$ (except in dimension $2$, where we can take powers of the Cauchy-Riemann operator).