Is this system an FIR filter or an IIR filter?
It is an IIR filter.  I think you misunderstand what is meant by "finite impulse response".  It does not mean finite amplitude, it means finite time- i.e. that the impulse response eventually becomes zeros.

The impulse response of the filter is a step function, which is infinite in time.

x[-1] = 0
x[ 0] = 1
x[ 1] = 0 ...

y[-1] = 0
y[ 0] = 1
y[ 1] = 1
y[ 2] = 1 ...