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Based on this answer, I should be able to figure out this question, if I know the distance between Betelgeuse and its nearest neighbor. So what is Betelgeuse's nearest neighbor?
I do mean the nearest neighbor in 3D space, not in the plane of the sky. Since Betelgeuse is fairly close to the Sun, I'm guessing we should know its neighborhood with a pretty good precision.
$$ \begin{array}{|c|c|c|} \hline \textbf{Star} & \textbf{Magnitude} & \textbf{Distance (ly)} \\ \hline \text{Betelgeuse} & \text{0.45} & \text{0.00} \\ \hline \text{HIP27648} & \text{8.24} & \text{17.82} \\ \hline \text{HIP28478} & \text{8.42} & \text{19.03} \\ \hline \text{HIP27573} & \text{7.41} & \text{24.27} \\ \hline \text{HIP28175} & \text{9.1} & \text{39.34} \\ \hline \text{HIP26887} & \text{7.28} & \text{39.34} \\ \hline \text{HIP29261} & \text{7.66} & \text{41.42} \\ \hline \text{HIP26914} & \text{9.6} & \text{41.87} \\ \hline \text{HIP27513} & \text{8.28} & \text{42.65} \\ \hline \text{HIP27386} & \text{5.26} & \text{44.20} \\ \hline \text{HIP28261} & \text{8.45} & \text{46.08} \\ \hline \text{HIP28686} & \text{6.09} & \text{46.73} \\ \hline \text{HIP27004} & \text{7.89} & \text{47.16} \\ \hline \text{HIP29537} & \text{7.82} & \text{49.28} \\ \hline \text{HIP27324} & \text{9.22} & \text{49.88} \\ \hline \text{HIP27932} & \text{8.26} & \text{50.41} \\ \hline \text{HIP29868} & \text{8.92} & \text{50.69} \\ \hline \text{HIP28397} & \text{8.24} & \text{50.95} \\ \hline \text{HIP29236} & \text{6.57} & \text{52.93} \\ \hline \text{HIP27902} & \text{6.3} & \text{52.98} \\ \hline \text{HIP28232} & \text{6.22} & \text{54.81} \\ \hline \text{HIP27315} & \text{7.14} & \text{55.28} \\ \hline \text{HIP27751} & \text{6.9} & \text{55.37} \\ \hline \text{HIP30124} & \text{9.75} & \text{56.46} \\ \hline \text{HIP29685} & \text{7.69} & \text{56.79} \\ \hline \text{HIP27412} & \text{9.31} & \text{58.80} \\ \hline \text{HIP27153} & \text{7.46} & \text{60.13} \\ \hline \text{HIP28202} & \text{7.97} & \text{60.42} \\ \hline \text{HIP27919} & \text{7.15} & \text{60.85} \\ \hline \text{HIP28056} & \text{7.17} & \text{60.89} \\ \hline \text{HIP29381} & \text{8.88} & \text{61.26} \\ \hline \text{HIP26986} & \text{9.17} & \text{61.26} \\ \hline \text{HIP29630} & \text{8.15} & \text{62.56} \\ \hline \text{HIP29590} & \text{7.82} & \text{64.37} \\ \hline \text{HIP30120} & \text{6.76} & \text{64.57} \\ \hline \text{HIP26795} & \text{6.82} & \text{66.03} \\ \hline \text{HIP26655} & \text{7.09} & \text{66.99} \\ \hline \text{HIP26615} & \text{9.13} & \text{67.50} \\ \hline \text{HIP27895} & \text{7.04} & \text{68.80} \\ \hline \text{HIP27309} & \text{9.7} & \text{69.04} \\ \hline \text{HIP28323} & \text{8.16} & \text{69.12} \\ \hline \text{HIP28171} & \text{8.68} & \text{69.91} \\ \hline \text{HIP25767} & \text{8.93} & \text{70.26} \\ \hline \text{HIP25698} & \text{8.74} & \text{70.31} \\ \hline \text{HIP26107} & \text{7.83} & \text{71.05} \\ \hline \text{NuOri} & \text{4.42} & \text{72.70} \\ \hline \text{HIP27284} & \text{8.72} & \text{73.29} \\ \hline \text{HIP27968} & \text{8.42} & \text{73.36} \\ \hline \text{HIP29599} & \text{7.96} & \text{73.41} \\ \hline \text{HIP25424} & \text{8.63} & \text{73.44} \\ \hline \end{array} $$
Subject to important caveats below, the table above lists the 50 stars closest to Betelgeuse including Betelgeuse itself:
I ignore my own advice and used HYG data to create this list.
The repl.it code: https://repl.it/@barrycarter/Twitch-BeetleJuice
I hope to turn this into a webapp later; for now, by changing the var fixedStar = stars[27919]; line in script.js, you can run this for any star in the HYG catalog (be sure to use the HYG id). You can convert to LaTeX using https://barrycarter.github.io/pages/CSV2LATEX/
As others have noted, the distance to Betelgeuse is not accurately know: "https://en.wikipedia.org/wiki/Betelgeuse#Distance_measurements". Of course, the same may apply to other stars in the HYG catalog, including those I have listed above. Therefore, the computed distance between Betelgeuse and another star may be way off.
I'm hoping to get a "better" list by using GAIA DR2 data, and will amend this answer when I do. Of course, while GAIA DR2 lists more stars, the distance measurements can be even further off.
Barry Carter's nice piece of work contains 50 stars within $\sim 74$ light years of Betelegeuse.
Leaving aside the question marks over exactly where Betelegeuse is (in terms of distance) and the parallax uncertainties of the 50 stars listed, there is also the problem that the faintest stars in the HYG catalogue are based on the Hipparcos catalogue, which is only complete to something like 8th magnitude, though it does contain some stars down to about 10th mag. If we assume that Betelgeuse is at around 700 light years (214 pc), then a magnitude of 10, corresponds to an absolute magnitude of about 3.3. These stars are very luminous indeed, about 3 times as luminous as the Sun or more, so either upper main sequence stars are giants.
Upper main sequence stars and giants form a tiny fraction of the local stellar population (and one presumes the locale around Betelgeuse since it is only 30 pc below the Galactic plane) - see for example this plot from Jeffries & Elliott (2003), which shows an absolute magnitude vs colour (HR) diagram for 1000 stars within $\sim 15$ pc of the Sun. As you can see, objects with $M_V<3.3$ are rare, perhaps $\sim 2$% of the population.
So the point of this is, that assuming Betelegeuse isn't sitting in a void, or in a cluster (which as far as we know, it isn't) then the stellar density around it is at least (and I say at least, because there is no way that the catalogue I used to produce the picture above is complete beyond spectral type M3-M4) is at least 0.07 stars per cubic parsec, or 0.002 stars per cubic light year.
If that is the case, then there should be at least 3400 stars within 74 light years of Betelgeuse and finally (the whole point of this "answer"), the nearest star to Betelgeuse is likely to be separated from it by $1/(0.002)^{1/3} = 7.9$ light years.
Unfortunately, it is an "answer" because we are no closer to knowing which star that is! Gaia DR2 (and DR3 later in 2020) would help in completing the census, since it has (or will have) parallaxes for stars down to about $V\sim 19$ (absolute magnitude of about 12.3 at the distance of Betelgeuse) and should then include about 60-70% of stars in its vicinity. However, Gaia will not (I think) yield an accurate parallax for Betelgeuse because it is too bright(!), so we will be no further forward in answering the question.
