Guess Number

My tech-lover son has been attracted by the exciting game "Guess Number". In this game, the player must find a hidden positive integer number by at most $T$ guesses (or turns). The parameter $T$ together with a health parameter $H$ is determined at the beginning of the game. In each turn, the player must enter a number. If the number is equal to the hidden number, he wins provided that H \geq 0. If the number is bigger than the hidden number, $H$ is decreased by $1$ unit of health. Otherwise, $H$ remains unchanged. When $H$ becomes negative or $T$ reaches $0$, the player definitely loses. The player can see the remaining turns and units of health after each turn. Although the game seems to be constructive, but something makes me suspicious! my son always wins. He claims he has a searching algorithm to find the hidden number, but I can't believe him as for the given $H$ and $T$ there must be a big number which is not guessable by any searching algorithm. To prove my son claim is wrong, I kindly ask you to help me find the smallest $M$ for which at least a number from $1$ through $M$ as the hidden number can't be guessed for the given $T$ and $H$.

For example, there is not any algorithm for finding all positive integers not greater than M = 3 by $2$ turns and $0$ units of health.