One option that doesn't seem to be mentioned is the used of beamed energy such as laser (visible or infra-red spectrum) which provide energy directly to a rocket.
This might be used in a secondary process (e.g., ion or plasma generators) or directly (heating atmosphere and/or fuel) to generate thrust.
The advantage is that the power source is on the ground, and need not be lofted, which removes part of the rocket-equation limit. It's still required to source or carry reaction mass, and I'd suggest that at least a fair portion of that be obtained within the atmosphere.
I don't know what a launch trajectory would look like, though I suspect something which went relatively slowly vertical (to minimise low-elevation drag), then began a hybrid lifting-ballistic flight at the highest possible levels of the atmosphere, powered by a planet-ringing set of laser stations, and acquiring reaction mass from the atmosphere itself, might be within the realm of reason?
It also strikes me that a world with sufficient mass would tend to retain hydrogen gas itself (though that would still likely react with oxygen to form water vapour), but at higher elevations there might be a significant differential fraction of H2 to other atmospheric components. Root mean squared velocity of H2 at 27 C (300 K) is about 7,000 kph (~4,300 mph).[1]
That's already less than Earth's escape velocity, so the problem is the molecules which have higher velocity that "boil off" into space.[2] I don't have the chops to compute this.
But a laser-pumped mesospheric hydrogen ramjet rocket might be able to take advantage of highly-energised (heated or ionised) hydrogen to gain escape velocity on even a significantly larger Super-Earth.
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Notes:
1. https://chem.libretexts.org/Bookshelves/General_Chemistry/Ch...>
2. Earth has lost roughly 25% of its primordial hydrogen (and water) by this mechanism. https://sciencenordic.com/chemistry-climate-denmark/the-eart...>