Does $y[n] = x[n] \star (u[n]-u[n-2])$ have memory or is it memoryless?
The system $$y[n] = x[n] \star (u[n]-u[n-2])$$ where $u[n]$ is the unit step function, has memory.

Indeed the system is equivalent to $$y[n] = x[n] \star ( \delta[n] + \delta[n-1] ) \implies y[n] = x[n] + x[n-1]$$ and as it's clear from the given I/O relationship, the current value of the output $y[n]$, depends on the values input $x[n]$ at other times namely $x[n-1]$ (one step past value); hence the system has memory.