EU regulators ruled Apple's controversial tax deal was illegal, and is demanding the record penalty.
The tech giant says it has been singled out and was "a convenient target".
Ireland is also contesting the decision, claiming EU regulators were interfering with national sovereignty.
Apple's European headquarters are located in Ireland - where the standard rate of corporate tax is 12.5%.
But in August, the commission said Ireland had enabled the company to pay substantially less than other businesses, in effect paying a rate of no more than 1%.
Ireland's finance ministry said in a strongly-worded statement on Monday that the European Commission had "misunderstood the relevant facts and Irish law".
"Ireland did not give favourable tax treatment to Apple - the full amount of tax was paid in this case and no state aid was provided," it said. "Ireland does not do deals with taxpayers."
And Apple's general counsel Bruce Sewell told Reuters that the commission had disregarded tax experts brought in by Irish authorities.
"Apple is not an outlier in any sense that matters to the law. Apple is a convenient target because it generates lots of headlines," Mr Sewell said.
Even if Apple lost its appeal, the record tax bill should not be a problem for iPhone maker which saw a net profit of $53bn in the 2015 financial year.
Apple is not the only company that has been targeted for securing favourable tax deals in the European Union.
Last year, the commission told the Netherlands to recover as much as €30m (£25.6m) from Starbucks, while Luxembourg was ordered to claw back a similar amount from Fiat.
Apple and Ireland
€13bn demanded in back taxes. Equal to:
ALL of Ireland's healthcare budget
66% of its social welfare bill
15 million iPhones
27% of Apple's 2015 profit
[Apple, HSE, Dept of Social Protection]
Source: BBC Business News