I now have
Four double checks by promotion in a row.
FEN: B1nQ2B1/1P1PPP2/2k5/8/8/8/8/7K w - -
Game: 1. b8=N+ Kd6 2. dxc8=N+ Ke6 3. f8=N+ Kf6 4. e8=N+
Proof that it is optimal: UNDER EDIT
A double check can occur either by a classic discovery, or by an en-passant capture. The latter is impossible here so it has to be a discovery. There are three cases:
Type A: Black King in the 8th rank right in front of the pawn to be promoted, promotion = Q or R, with another Q or R in the same column as the King.
Type B: Black King in the 7th rank, checked by Rook or Queen also on the 7th rank; all four types of promotions can occur.
Type C: Black King in the 6th rank, attacking-close to the pawn to be promoted. Promotion = Q or R or N, while the other checking piece can be anything.
Now here is why there can't be too many moves:
Fact 1: there can be only one type A in the sequence. Because (long story short) the King has to leave the 8th rank and it can never come back.
Fact 2: there can be only one type B in the sequence. Because (long story short) the King has to leave the 7th rank and it can never come back.
Therefore the only possible sequences are (possibly truncated) ABCCC... and ...CCCBA.