The quickest mate in n Obviously inspired by the analogous selfmate  question. (Helpmate would also be possible in principle, but looks a bit silly.)
Thus, 1.f3 e5 2.g4 allows Black a #1, and 1.f3 e5  2.h3 a #2. 1.f4 e5 2.Nf3 Be7 3.h3 is a #3. This should be the quickest games.
Find a shortest helpplay to a position where one side has a forced #4 (and of course it must be unique and not be shortenable), for starters.
A mere 3 moves are needed to reach a position in which White has a forced #4.
[FEN &quot;&quot;]

1. e3 b6 2. Bc4 Bb7 3. Qf3 d6 {Now the helping play is over.}  4. Qxf7+ Kd7 5. Qf5+ e6 6. Bxe6+ Kc6 (6... Ke7 7. Qf7#) 7. Qd5#

However, longer mates are also possible to produce in just three moves.
Here is a #5.
[FEN &quot;&quot;]

1. e4 e5 2. Bc4 f5 3. Qe2 h6 {Now the helping play is over.} 4. Qh5+ g6 5. Qxg6+ Ke7 6. Qf7+ Kd6 7. Qd5+ Ke7 8. Qxe5#

Even a #6 can be done.
[FEN &quot;&quot;]

1. e3 d6 2. Bc4 b6 3. Qf3 b5 {Now the helping play is over.} 4. Qxf7+ Kd7 5. Qf5+ e6 6. Qxe6+ Kc6 7. Qd5+ Kd7 8. Qf5+ Kc6 9. Qxb5#


Addendum 5/22/2021-A is #7 indeed possible, albeit in 4 moves.
[FEN &quot;&quot;]

1. e4 e5 2. Bc4 d5 3. Qh5 Ke7 4. f4 Kf6 {Now the helping play is over.} 5. Qg5+ Ke6 6. Qxe5+ Kd7 7. Qxd5+ Ke7 8. Qxf7+ Kd6 9. Qd5+ Ke7 10. Qe5+ Be6 (10... Kd7 11. Qe6#) 11. Qxe6#

For an #8, 5 moves is my best result so far.
[FEN &quot;&quot;]

1. e3 e5 2. Qh5 Ke7 3. Bc4 Kf6 4. c3 g6 5. d3 a6 {Now the helping play is over.} 6. Qf3+ Kg7 7. Qxf7+ Kh6 8. e4+ Qg5 9. Bxg5+ (9... Kxg5 10. h4+ Kh6 11. Qxf8+ Kh5 12. g4+ Kxg4 13. Qf3#) Kh5 10. g4+ Kxg4 11. h4 Nf6 12. Qxf6 Kh5 13. Qf3#