As a child my Spirograph was definitely a favourite toy so I was delighted to find this digital version, Inspirograph by Nathan Friend. Try altering the gears so that the fixed and rotating gear are the same size, or make one size a factor of the other, make the two sizes have a common factor, or not! Investigate. You can change the colours too and create a work of Art!
The Nrich problem ‘Making Maths: Planet Paths‘ challenges students to draw some planet paths using a Spirograph. In case there is no Spirograph to hand they give instructions for making a simple one.
Alternatively try an online version. Try Spirograph on the Desmos Graphing calculator.
From mathplayground.com comes this very clear and easy to use applet (the moving circle is outside the fixed circle).
For GeoGebra fans there are various applets available, including this which allows colour changes.