Immigrating to the us without a visa but entering legally?

Without a visa of some sort it is impossible to enter legally

Technically that's not "immigration" but rather just overstaying. It puts you in a slightly different legal category from someone who entered without inspection, but doesn't make you any less deportable. No, you can't overstay and then be eligible for a green card. If that were possible the US wouldn't currently be home to some 10-30 million unlawfully present individuals. 

No.

Also, if that is their plan, then they cannot enter legally.

It is not legal to enter without a visa, unless you are either (a) applying for political asylum, or (b) a citizen of a visa waiver program country who plans to leave within a certain number of days.

They would not be admitted.  Entering without either a visa or a plan to leave would make them ineligible.

NO.....................

NOT possible. NO way to legally enter the US without your passport PLUS visa or visa waiver (ESTA). With visa waiver, for citizens of eligible countries, there is NO visa extension, NO adjustment of status, NO exceptions but to return to country of citizenship within 90 days. The provisions of visa waivers are strict, and extensions, adjustment of status, etc. are strictly & specifically prohibited. The only way to enter US with eligibility to obtain a green card (legal permanent residency status) is to have a currently valid immigration visa. Those who enter legally with passport plus temporary visa (e.g. tourist, student, employment) are on temporary visas and are NOT eligible for green cards.