As a child, a favourite toy was my Spirograph with which I could generate endless pretty curves! Speaking of my favourite toy, there is a wonderful electronic version available.
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.
Owen Elton has written an excellent Spirograph Autograph activity (see Simon’s comment below) and also available on the Autograph player a very impressive 3D Spirograph!
On GeoGebra, there are various applets available, for example: