John Conway's Game of Life (1970). Every cell is alive or dead, and on each tick: a live cell with 2 or 3 live neighbours survives, a dead cell with exactly 3 comes alive, and everything else dies. That's the whole rule set, and it's Turing complete.

Click or drag to draw cells. Choose a pattern to stamp it where you click (pick "stamp a pattern…" again to go back to drawing). space plays or pauses, . steps one generation. The grid wraps around like a torus. Brand-new cells flash white, then cool through blue to purple as they age.

The starting pattern is the Gosper glider gun (1970), the first known pattern that grows forever.