Interpreting the question the same way as AplanisTophet, I'm pretty sure the answer is no more than two.

[1, 2] are both divisible by 1.

[2, 4] $\in$ [1..4] are divisible by 1 and 2.

[6] $\in$ [1..6] is divisible by 1, 2, and 3.

No values in [1..8] are divisible by 1, 2, 3, and 4. (LCM 12)

No values in [1..10] are divisible by 1, 2, 3, 4, and 5. (LCM 60)

No values in [1..12] are divisible by 1, 2, 3, 4, 5, and 6. (LCM 60)

No values in [1..14] are divisible by {1..7}. (LCM 420)

No values in [1..16] are divisible by {1..8}. (LCM 840)

*&c. &c.*