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Here is a list of H-Beta targets, most of which are faint:
1. IC 434 (HORSEHEAD NEBULA)
2. NGC 1499 (CALIFORNIA NEBULA, naked eye and RFT)
3. M43 (part of the Great Orion Nebula)
4. IC 5146 (COCOON NEBULA in Cygnus)
5. M20 (TRIFID NEBULA, main section)
6. NGC 2327 (diffuse nebula in Monoceros)
7. IC 405 (the FLAMING STAR NEBULA in Auriga)
8. IC 417 (diffuse Nebula in Auriga)
9. IC 1283 (diffuse Nebula in Sagittarius)
10. IC 1318 GAMMA CYGNI NEBULA (diffuse nebula in Cygnus)
11. IC 2177: SEAGULL NEBULA (Diffuse Nebula, Monoceros)
12. IC 5076 (diffuse nebula, Cygnus)
13. PK64+5.1 "CAMPBELL'S HYDROGEN STAR" Cygnus (PNG 64.7+5.0)
14. Sh2-157a (small round nebula inside larger Sh2-157, Cassiopeia)
15. Sh2-235 (diffuse nebula in Auriga).
16. Sh2-276 "BARNARD'S LOOP" (diffuse nebula in Orion, naked eye)
17. IC 2162 (diffuse nebula in northern Orion)
18 Sh2-254 (diffuse nebula in northern Orion near IC 2162)
19. Sh2-256-7 (diffuse nebula in northern Orion near IC 2162)
20. vdB93 (Gum-1) (diffuse nebula in Monoceros near IC 2177)
21. Lambda Orionis nebular complex (very large, naked-eye)
22. Sh2-273 "Cone" Nebula portion south of cluster NGC 2264
A list of objects at the limit of 12.5" and easier with larger apertures would number in the thousands, so not practical to answer.
There is a galaxy cluster in Corona Borealis at a bit over a billion light years that would be a good candidate, since larger apertures will see more galaxies.
There are a lot of galaxy clusters in the Southern sky that would qualify as well, like Abell 838 in Hydra at 675 million light years, or Abell 2448 in Aquarius at a distance of 1.1 billion light years.
There are many many such across the entire sky.
The limit depends as much on darkness of sky as it does on aperture.
15th magnitude galaxies are within reach in a 12.5" in dark skies, but 13th magnitude galaxies are tough in light polluted suburbs.
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