This is headcanon, but I always thought that Lavos somehow survived the constant death and rebirth of the universe. The universe is created via Big Bang, then eventually reaches a Cold Death and collapses in on itself, only to condense and recreate the Big Bang all over again. A never-end cycle of universes.
Lavos would somehow be able to survive and flow from one universe to another, each time trying to harness that universe's strongest to continue to evolve itself... and would in turn guide the new universe's evolution to try and make it stronger than the last.
One of my head canon Chrono fangame ideas (never to be realized) dealt with this. But whereas time is malleable in this Chronoverse and can easily be changed via time travel, the new universe would actually be fatalistic time travel, in that attempts to change time end up only fulfilling events that occur. The late-game quest would be to break the chains of fate that governs this version of time travel so that people aren't bound to this form of time travel. It would have also hinted at Lavos, with Lavos ending up being the big villain in a potential sequel.
Alas! Nothing worse than unrealized story ideas.