Here’s something I didn’t know. There exists nonlinear (but algebraic) ordinary differential equations such that solutions to that differential equation are dense in the space of continuous functions. These are known as universal differential equations. An explicit construction of one is given in this preprint by Keith Briggs. If I understand the construction correctly, the trick seems to be that the nonlinearity gives you branch points where you have a choice for the direction in the solution. This allows you to paste together solutions in enough ways that you can achieve density.