Do larger numbers have more prime factors than smaller ones?

No.

Look at the following examples:

The prime factors of 36 are 2, 2, 3, and 3.

The prime factors of 62 are 2 and 31.

The prime factors of 155 are 5 and 31.

The prime factors of 168 are 2, 2, 2, 3, and 7.

The prime factors of 194 are 2 and 97.

The prime factor of 197 is 197.