Could a Trappist-like planetary system exist around a sun like ours? So, a couple of years ago, the TRAPPIST sun was discovered to have 7 different earth-like planets orbiting it – three of them in the habitable zone. Nothing like this has ever been seen before, or found again since. And the main reason it's possible at all is because of the orbital resonance of the planets – they orbit in a pattern which causes the system to be stable long-term. And this whole system of planets take place closer to the TRAPPIST star than Mercury orbits our sun.
This is interesting for a lot of reasons, but one that appealed to me was that so many planets could fit – and be stable – in such a small space. I was under the impression that you couldn't fit that many planets so close together.
(Another interesting fact: The distance between TRAPPIST-1B and TRAPPIST-1H is less than the distance between the two edges of our sun's Goldilocks zone.)
So my question is this: could such a system be stable around a larger sun – a sun like ours? And could such a system work with the planets further out – in the habitable zone? Would it be stable or am I missing something?
There are a couple of systems around solar-mass stars that follow the pattern of tightly-packed planets orbiting close to the star. These include:

Kepler-11 (0.961 solar masses), a system of six low-density sub-Neptune planets orbiting within 0.5 au of the star
HD 10180 (1.06 solar masses), a system of six Neptune-size planets of which four are within 0.5 au of the star.
Kepler-90 (KOI-351) (1.2 solar masses), a system with eight planets within 1 au of the star ranging in size from Earth-sized to Jupiter-sized.

In the case of HD 10180, planet g is located in the liquid water zone, though its mass indicates that it is an ice or gas giant planet and thus not habitable. Kepler-90's outermost planet is located at 1 au, but the star is more luminous than our Sun so it might be too hot to support liquid water and is in any case a Jupiter-sized planet. So systems like TRAPPIST-1 with packed inner planetary systems clearly exist around stars like the Sun, although the difficulty of finding small planets at sufficient distance from the star means that so far we haven't found such systems with terrestrial planets in the liquid water zone.
The question of how many planets you could pack into the habitable zone has been investigated by Kane et al. (2020), &quot;Dynamical Packing in the Habitable Zone: The Case of Beta CVn&quot;, where they investigate the maximum number of planets that can dynamically fit in the habitable zone by spectral type, and also consider in detail the specific case of the G-dwarf Beta Canum Venaticorum. From the paper:

Our analysis
presented here demonstrates that in fact the dynamical limitations to the packing of HZ terrestrial planets
is ∼5 planets for most spectral types, and ∼6 planets
for stellar masses $\gtrsim 0.7 M_\odot$. Packing 7 planets in the
HZ is possible within certain specific stellar mass and
architecture regimes, but becomes vulnerable to MMR
perturbations that compromise the dynamical stability
of such configurations.

(MMR = mean motion resonance)
Whether real systems actually reach such a dense packing in the habitable zone is unknown, and they note some caveats to their analysis:

There are some caveats to the analysis presented here
that are worth repeating. First, Section 2.1 describes
the use of 3 Gyr isochrones, which are adopted in order
to include stars at the higher mass end of the stellar
mass range. However, the star will evolve with time,
as will the HZ (Gallet et al. 2017), resulting in an increase in both the width and orbital distance of the
HZ. Subsequently, planets located at the inner edge of
the HZ at 3 Gyr will transition into the Venus Zone
(Kane et al. 2014) where runaway greenhouse scenarios may become dominant. Second, our simulations account for Earth-mass planets and circular orbits. Higher
masses and non-zero eccentricities will reduce orbital
stability within the HZ, but lower mass planets may allow for additional planets, with the exception of strong
resonance regions.

(HZ = habitable zone)