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Jupiter's mass is about the maximum a planet can be before it starts to fuse hydrogen in its core and undergo a massive transformation.

Total mass of all solar system objects in question is: here.

Jupiter is 1 Jupiter large, while the summation of all else is 0.405 Jupiter large.

Given the mass of Jupiter and the mass of all other planets, if Jupiter were to somehow collide and "absorb" all mass from every planet in the current solar system, would it turn into a star?

I guess that you can distill this question down to:

If Jupiter were 1.405x more massive, would it be a star?

Note: Still kind of new to these concepts, please explain misconceptions I may have.

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No, hydrogen burning doesn't start until 82 $m_{\rm Jup}$ and deuterium burning starts at 13 $m_{\rm Jup}$, so evidently 1.4 $m_{\rm Jup}$ is not enough for this experiment.

Your misconception is that

Jupiter's mass is about the maximum a planet can be before it starts to fuse hydrogen in its core and undergo a massive transformation.

which is simply not true.

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  • $\begingroup$ Thank you for clearing that up in terms of Jupiter mass :). $\endgroup$ – Magic Octopus Urn Aug 27 at 17:24
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    $\begingroup$ Jupiter may not have the maximal mass far a planet, but is very close to the maximal size of a planet. Adding more gas, without starting fusion, would (almost) only increases density not radius. This might have been what you have heard and what led to some confusion. $\endgroup$ – SpaceBread Aug 27 at 21:33
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The above answer handles the question as you posed it, but perhaps it should also be addressed the way you phrased the question in the title. Are you imagining that degeneracy pressure is associated with being a star? Degeneracy pressure is achieved simply by waiting long enough for a gas to lose enough heat. Jupiter is already pretty close to old enough to be supported by degeneracy pressure, all without ever getting anywhere close to being a star.

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