UPDATE #2: Now I have discovered a case of 16 ply for a sequence of only one legal moves that results in checkmate! It can be found here in a PDF of Feenschach, the 1980 edition.
[Title "Bernd Schwarzkopf nach Karl Scherer (Urdruck), Mate In 8"]
[FEN "1KN4r/QRRRRRRr/kq6/p1q1R3/PP1q1b2/4q3/5q2/1r4q1 b KQkq - 0 1"]
1... Qxa7+ 2. Rxa7+ Qxa7+ 3. Rxa7+ Qxa7+ 4. Rxa7+ Qxa7+ 5. Rxa7+ Qxa7+ 6. Rxa7+ Qxa7+ 7. Rxa7+ Rxa7 8. b5+ Rxb5+ 9. axb5#
UPDATE: Sorry Professor Elkies, but it looks like someone beat your 14 with a 15 two years before you in the chess.com forums!
I found this the following, and amazing, sequence here in the chess.com forums. This was created by a person with the username of shoopi
[Title "shoopi, 2013"]
[FEN "rk1KB3/b1q1QQQR/bnqQ1q2/1Rq1Q1q1/2q1rQ1q/6B1/6QQ/8 w - - 0 1"]
1. Qxc7+ Qxc7+ 2. Qxc7+ Qxc7+ 3. Qxc7+ Qxc7+ 4. Bxc7+ Kb7+ 5. Bb8+ Qxe7+ 6. Qxe7+ Qxe7+ 7. Qxe7+ Qxe7+ 8. Rxe7#
If you are wondering if this is legal, let's look at the position. Black has 6 promoted and white has 7. Using a trick that I leaned from Professor Elkies (This was when I read from Tim Krabbe'a diary entry of his evidence for the legality of his 19 ply no-brainer puzzle. Thanks for that Mr. Elkies!), here is a short proof game of legality that I made myself.
1. d4 c5 2. b4 a5 3. dxc5 axb4 4. Nc3 e6 5. Nd5 exd5 6. g4 h5 7. gxh5 Nf6 8. f3 Ne4 9. fxe4
And now black and white are capable of promoting to all of the queens seen in the diagram.
Here is another 14-mover that ties with Noam D. Elkies’s composition of 14 half-moves. Of course, it is based off of the position given by Hauke Reddmann in his answer to this similar question here on CSE.
[Title "Rewan Demontay, 2019"]
[FEN "1Qb5/1pQp4/1P1Q4/3PQ2p/5Q1P/6QK/Rrrrrrrq/R4nk1 w - - 0 1"]
1. Qxh2+ Rxh2+ 2. Qxh2+ Rxh2+ 3. Qxh2+ Rxh2+ 4. Qxh2+ Rxh2+ 5. Qxh2+ Rxh2+ 6. Qxh2+ Rxh2+ 7. Rxh2 d6#
It may not look like a legal position, but I assure you that it is legal. A proof game made by SpiderUnicorn can be found here in the chess.com forums. (I basically turned it into a puzzle because I had NO idea myself whether or not legal position!)
I believe that this shows that 14 is the definitive maximum for this kind of problem.
If you want to know the record for one legal move regarding discovered checks, I can give you this.
I found this funny example on Tim Krabe’s website (Journal Entry #265.)
He gives this series of 7 mutual discovered checks. All of the moves, minus the first, are forced, which is what makes it unique.
[Title "V. Korolkov, 1940"]
[FEN "6B1/5Nb1/3p4/q2krP1R/Nn2p3/pPKnr3/Q1PB4/3R4 w - - 0 1"]
1. Nd8+ Re6+ 2. f6+ Ne5+ 3. Bxe3+ Nd3+ 4. b4#