We have not yet discovered any exomoons. Many inevitably exist, as all planets in our system except Mercury and Venus have moons. Our instruments for detecting them elsewhere in the universe are too crude yet. Gas giant exomoons will likely be found in the future and will not be much smaller than the exoplanet they orbit.
As the above image shows, gas giant planets don't get much more significant in diameter with the increase in mass due to compression from high gravity. Brown dwarf stars start from 13 times the mass of Jupiter, while lowest-mass red dwarf stars are at least 80 times as massive as Jupiter, yet they aren't that much bigger in diameter. We found some outliers in this regard. Some gas giant exoplanets orbit their stars so close that they get puffed up from being fried by high temperatures.
A very massive gas-giant exoplanet that is nearly 13 times the mass of Jupiter can potentially have moons that are Neptune, Saturn, or even Jupiter-like, and they would be gas-giant moons.
In the past, we had doubts about whether such moons could form, but in October 2023, the James Webb Space Telescope discovered many Jupiter-like double rogue exoplanets floating in space without stars in the Orion Nebula. The current leading theory is that they formed in typical star systems and were ejected. Some planets can end up jettisoned from systems during planetary formation, collide, or fall into the star and be destroyed before the orbits settle on stable configurations.
If such double Jupiter-like planets indeed formed in star systems and not directly in space like suns form, then we will find double gas giant planets and massive gas giant planets with gas giant Jupiter- or Saturn-size exomoons in the future.
In our system, Titan, the moon of Saturn, is larger than the planet Mercury but less massive and has a thicker atmosphere than Earth. It is the closest we get to a gas giant moon in our backyard.