This work addresses the challenge of patrolling regular grid graphs of any dimension using a single mobile agent with minimal memory and limited sensing range. We show that it is impossible to patrol some grid graphs with $0$ bits of memory, regardless of sensing range, and give an exact characterization of those grid graphs that can be patrolled with $0$ bits of memory and sensing range $V$. On the other hand, we show that an algorithm exists using $1$ bit of memory and $V=1$ that patrols any $d$-dimensional grid graph. This result is surprising given that the agent must be able to move in $2d$ distinct directions to patrol, while $1$ bit of memory allows specifying only two directions per sensory input. Our $1$-bit patrolling algorithm handles this by carefully exploiting a small state-space to access all the needed directions while avoiding getting stuck. Overall, our results give concrete evidence that extremely little memory is needed for patrolling highly regular environments like grid graphs compared to arbitrary graphs. The techniques we use, such as partitioning the environment into sensing regions and exploiting distinct coordinates resulting from higher-dimensionality, may be applicable to analyzing the space complexity of patrolling in other types of regular environments as well.