At one light-year per second, you'd be traveling 31,536,000 times the speed of light. Yet if you tried to fly "out" of the universe, you would eventually just arrive right back at Earth.
Even at this incomprehensible velocity, outpacing cosmic expansion with a magic warp drive, there is simply no "out" to escape into. First, let's look at the sheer scale of the space around us. The observable universe—the spherical region from which light has had time to reach Earth since the Big Bang—has a radius of about 46.5 billion light-years. If you point your ship in one direction and hit the gas at one light-year per second, it would take exactly 46.5 billion seconds to reach that cosmic horizon.
That equates to roughly 1,474 years. So even at more than 31 million times the speed of light, it takes nearly a millennium and a half of non-stop travel just to reach the edge of what we can currently see.
Crossing that 46.5-billion-light-year boundary does not mean you leave the universe. The observable universe is just a subjective bubble centered on the observer. When you cross that threshold, you simply enter a region of space that was previously invisible from Earth. If you looked around from your new vantage point, you would find yourself at the center of a brand new observable bubble, surrounded by billions of unfamiliar galaxies stretching out in every direction.
Cosmologists generally agree that the universe lacks a physical edge or boundary wall. It is either infinite, extending endlessly with the same distribution of stars and galaxies, or it is finite but unbounded. The latter scenario means the fabric of spacetime curves back on itself in higher dimensions—much like how the two-dimensional surface of the Earth wraps around a three-dimensional sphere.
If the universe is finite, keeping the throttle pinned at one light-year per second would never punch you through an outer shell. Instead, you would eventually circumnavigate the entire cosmos and arrive right back where you started, approaching Earth from the exact opposite direction you left.