score:12
To enter the US under the VWP, you will need an ESTA.
The ESTA form explicitly asks:
Have you ever been issued a passport or national identity card for travel by any other country?
(emphasis mine)
So, yes, you should indicate that you have had an Argentinian passport, even though it expired.
If you still have the passport, you can fill in all the relevant fields. If you don't have it, as explained in the relevant help section (accessible by hovering on the circled question mark):
If you have been issued one of these documents from another country, but you do not remember the passport number of national identification card number and the year of expiration, answer "UNKNOWN" in the passport number field and four zeros "0000" in the year of expiration field.
And no, there's no reason they would not allow you in because your other passport has expired or you no longer have it.