Infinite sequence of consecutive checks allowing more pieces (say 9+ queens) This is a task related to this question: Theoretical limit for the number of consecutive checks?.
By a sequence of consecutive check, what we mean is white giving check, black replying with a check, white replying to that with a check and so on. Note that we are talking about a theoretical situation so that 50 move rules or three-fold repetition are irrelevant.
In the linked question, he is asking for the longest sequence of consecutive checks with two conditions:

Initial position should be legal
Initial position should not contain promoted pieces.

Suppose we remove the 2nd condition, and relax the 1st one to allow more pieces than legally possible (say, 9+ white queens or 3+ white knights). With these relaxations, is it possible to have an infinite sequence of consecutive checks?
I think the answer to the above question is no (but no complete proof yet).
If such an infinite sequence is there, it should contain a loop for sure. Therefore, the sequence cannot contain pawn movements or captures.
A second question is to construct longest such sequence. Alexey Khanyan's record 54 with conditions 1 and 2 may be surpassable because we relaxed them. (By the way, even with conditions 1 and 2, Khanyan himself thinks that 54 can be improved).
PS: For this question, I am not interested in involving fairy chess pieces or chess variants.
No, it's impossible to have an infinite sequence of checks
Let's remember that there are 3 ways to parry a check :

Move the King to a safe square
Capture the enemy piece giving the check
Move a piece between the King and the enemy piece giving a check (except against a Knight check as it can jump over pieces)

2 and 3 aren't sustainable over an infinite number of moves, because the chessboard has finite space, only 64 squares and necessarily less pieces than there are squares. So the only way to get an infinite cycle of checks going would be for both Kings to escape checks with method 1, by continually moving them.
This also implies that the King movement gives check. Since Kings can't come into contact, that means their movement must deliver a discovery check from a long range piece. This in turn, implies that after each move, both Kings stay aligned over some axis (rank, file, or diagonal), otherwise there would no longer be any discovery checks for lack of an alignment.
Let's say we're in such a position and it's Black to move. To summarize the constraints, the Black King must :

Be on 1 of the 8 lines passing by the square where the White King is (otherwise it can't deliver a discovery check), and
Move to another of these 8 lines (otherwise White won't be able to deliver a discovery check), and
Follow normal rules of chess, e.g. move only one square in any direction and not on a square where it would be in check

Let's say the White King is on e4. The only way to move from one of the 8 lines passing by e4 to another by jumping only one square is to either go to or from one of the 4 squares diagonally adjacent to the White King : d5, d3, f3, f5. But because these squares are in contact with the White King, it's not possible for the Black King to perform such a move. It can't be on these squares because the position would be illegal, and it can't move there because the move is illegal. So it's impossible to follow these constraints.
Hence, from all the above, I'd conclude that under the normal rules of chess, it's impossible for both players to perform at all an infinite cycle of checks. This is also valid under the 2 conditions you mentioned in your question, as they don't affect the reasoning used here.