I have been in the United States since the 1st of May 2013 on a 90 day Visa Waiver Program ESTA. This means the ESTA expires by the end of July. We want to take our flight home to Australia on September 10th. Because we only technically need another 40 days of travel time in the Unites States, will it be possible for me to apply for a second ESTA whilst inside the U.S in order to stay on. How difficult is this/ is there anything I should know about second round ESTA applications like rules for leaving time inbetween ESTA grants. All ofour flights are paid for and I have an Australian citizen passport with intentions to totally return home permanently.
The duration of stay in the US is not a function of ESTA but a function of VWP which allows you as a citizen of Australia 90 days to stay in the US. You can apply for new ESTA 30 days before your current one expires, however, validity of ESTA is 2 years so for 2 years you can enter the US under Visa Waiver Program and each time stay for a total of 90 days.
Since you want to stay for 110 days you will need to obtain a standard B1/B2 visa unless you're planning to leave the country for a week or 2 and then come back. BTW, Mexico, Caribbean and Canada doesn't reset the count
The Visa Waiver Program only allows you to stay in the US for a maximum of 90 days. Once you are inside the US, there is no way to extend this (You could have applied for a visa before your stay commenced, but you can't do that once you're in the US).
Your only option is to leave the US, and return - but odds are that even this will cause issues. If you leave the US to a bordering country (Canada, Mexico) then the 90 day period will NOT be reset.
If you leave to any other country (eg, Australia, UK, etc) then the 90 day period will be reset - however it's likely that the customs official will be very interested in you having returned to the US so soon after your previous extended stay. Technically there is nothing wrong with returning immediately, but if the border official believes that your trip away from the US was done for the purpose of extending your time in the US (ie, a "visa run", which it fairly clearly was!) then it's very likely you will be denied entry.
The ESTA is not specifically relevant here. You require a valid, non-expired ESTA at the time you enter the country, but it doesn't in any way impact things like how long you can stay in the country.