In my logic, triple check not possible. Because I assume there is a move with two pieces at once needed. The only move I know is castling. The King can not give a check. So is triple check impossible?
Triple check is possible in xiangqi, aka chinese chess. There is even the possibility for quadruple check.
Unique to xiangqi is a triple check, which arises in three combinations. In the first case of a cannon, a chariot or soldier, and a horse, the horse moves to give check, uncovering a double check from the chariot and the cannon. In the second, rarer case of a chariot or soldier and two horses, the chariot moves to give check, uncovering a double check from the two horses. In the third case of two cannons and two horses, one cannon may uncover a double check from the horses and act as a screen for the other cannon. Quadruple check is also possible, arising with two horses, a chariot, and a cannon. Triple and quadruple check cannot be blocked.
Triple check is possible in the chess variant known as Cylinder Chess. In this position, 1.Ra8 discovers check from the queen and bishop, so it's triple or maybe quadruple check, depending on whether you count one or two checks from the rook, which is attacking the black king from two directions.
[FEN "4k3/8/8/8/R3K3/8/8/3Q1B2 w - - 0 1"] 1. Ra8+
Even if you could like take a pawn en passant and open up two lanes in 1 move, only 1 lane will be mathematically relevant to checking the king.
See, you can attack the king from a ton of directions, but all checking moves can mathematically only make 1 or 2 lanes/diagonal opened and thus relevant for delivering a check.
Chess notation for diagrams use + for a single check and ++ for a double check, but I've never seen a +++ character for a triple check. They simply don't exist.
The check itself defined as a player's king is under threat of capture by attacking piece(s) on next opponent's turn. According to standard chess rule, the pieces which can be counted as checking the king are pieces which able to move directly to the king's square.
These conditions counted as double check instead of triple check, assumed the pieces are attacking in one move:
1) Rook battery with queen/bishop/knight/pawn (second rook at the same file/rank not counted).
2) Queen-bishop battery with rook/knight/pawn (whoever supporting piece at the same diagonal not counted).
These conditions counted as single check instead of double:
1) Queen-rook battery (or two rook battery).
2) Queen-bishop battery.
Also, triple check is impossible in standard chess due to multiple checks must be executed in only one move, which allows maximum of 2 pieces capable to attack the opponent's king directly at the same time (attacking pieces are neither subject to be captured nor interposed by other opponent's pieces).