The standard definition is that $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the order of $a$ modulo $p$ is $p-1$.

Let me rephrase, to motivate my generalization: $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the linear recurrence defined by $x_0=1$, $x_n=ax_{n-1}$ for $n\geq1$ has maximum possible period in $\mathbb{Z}/p\mathbb{Z}$.

So, we could define $(a_1,\ldots,a_r)\in\mathbb{Z}^r$ to be an $r$-primitive root modulo $p$ when the order $r$ linear recurrence defined by $x_0=\cdots=x_{r-2}=0$, $x_{r-1}=1$, and $x_n=a_1x_{n-1}+\cdots+a_rx_{n-r}$ for $n\geq r$ has the maximum possible period (for such sequences) in $\mathbb{Z}/p\mathbb{Z}$.

Is anything known about the generalized version of primitive roots I've described? I chose my initial values $x_0=\cdots=x_{r-2}=0$, $x_{r-1}=1$ in analogy with the Fibonacci numbers, but is there a standard / most general choice? The starting values affect the period a lot, so it's important to be working with the "right" ones, I guess. What is the maximum achievable period for a linear recurrence in $\mathbb{Z}/p\mathbb{Z}$? An obvious upper bound is $p^r-1$, but it's not clear to me that this is the correct number to replace $p-1$ in the standard definition of primitive root. Is anything known in the direction of Artin's conjecture for "$r$-primitive roots", as I've called them?

Other thoughts: because there is no "formula" for the order of $a$ modulo $p$ (this would amount to a formula for the discrete logarithm), there certainly isn't a formula for the period of the linear recurrence defined by $(a_1,\ldots,a_r)$ modulo $p$. I tried to come up with one for a while before I realized this, so I just wanted to save anyone else the trouble.