Why don't reactions continue on forever?

Oscillating chemical reactions consume reactants. And like any other chemical reaction, eventually, one or more of the reactants becomes depleted and the reaction stops. Hence the reactions don't continue forever.

Explanation:

The oscillations come because the initial reaction generates intermediate compounds that are autocatalytic, speeding up the reaction. Crucially, these oscillations left to their own devices, the mixture slowly settles into an unchanging state. Oscillating chemical reactions consume reactants. And like any other chemical reaction, eventually, one or more of the reactants becomes depleted and the reaction stops (oscillating reactions do not recycle ingredients). So, regardless of what you would like to do, it can't be done because no such oscillating reaction exists.