To add to @Henning Makholm answer, your case is problematic in more ways than one and you're almost certainly guaranteed to be denied on more grounds than one.

Firstly you have demonstrated immigrant intent by your claim for asylum the first time around. That is a disqualifier for ESTA and non immigrant visas. At this point considering you fought this case for six years, nothing shows your intentions have changed and a reasonable immigration or consular officer (if you should apply for a visa) will deny your application.

Second you may even have committed immigration fraud and a zealous immigration officer could say you entered the previous time on ESTA knowing full well you were going to claim asylum. That's Preconceived intent, and that is basis for immigration fraud and that carries a permanent ban.

The fact that you did not accumulate more than one year or even one day of illegal presence in this case just means legally you do not have a bar and that you are not inadmissible. In practice however almost no consular officer or airport/border immigration officer will let you into the country knowing your immigration history. I mean it doesn't make sens for them to.

So even if you are able to argue that you never overstayed your ESTA because you filed asylum before your ESTA expired, your overall history makes you very very very unlikely to ever enter the USA on an non-immigrant visa.

PS: Even if you left the USA just one day after your asylum petition was denied, it means you had started accumulating unlawful presence from the day of that ruling.

To add to @Henning Makholm answer, your case is problematic in more ways than one and you're almost certainly guaranteed to be denied on more grounds than one.

Firstly you have demonstrated immigrant intent by your claim for asylum the first time around. That is a disqualifier for ESTA and non immigrant visas. At this point considering you fought this case for six years, nothing shows your intentions have changed and a reasonable immigration or consular officer (if you should apply for a visa) will deny your application.

Second you may even have committed immigration fraud and a zealous immigration officer could say you entered the previous time on ESTA knowing full well you were going to claim asylum. That's Preconceived intent, and that is basis for immigration fraud and that carries a permanent ban.

The fact that you did not accumulate more than one year or even one day of illegal presence (per your timeline) just means legally you do not have a bar and that you are not inadmissible. In practice however almost no consular officer or airport/border immigration officer will let you into the country knowing your immigration history. I mean it doesn't make sense for them to.

So even if you are able to argue that you never overstayed your ESTA because you filed asylum before your ESTA expired, your overall history makes you very very very unlikely to ever enter the USA on a non-immigrant visa.

PS: Even if you left the USA just one day after your asylum petition was denied, it means you had started accumulating unlawful presence from the day of that ruling unless you were specifically granted voluntary departure.