I was trying to figure out how many generations could theoretically exist at the same time, so I decided to first check Guinness. Hopefully I did all the math correctly.
I think OP made an error in stating you had to be 18. The oldest living person ever was 122 (as indicated by Blamemeta). As long as you are allowed a few pre-18 births, you can get to 7 generations). At 15 years old you can get to 8 generations.
You are missing things like this in your calculation.
Under current Pennsylvania law, teenagers aged 13, 14 and 15 may or may not be able to legally engage in sexual activity with partners who are less than 4 years older. Such partners could not be prosecuted under statutory rape laws, but may be liable for other offenses, even when the sexual activity is consensual.