You can apply after 5 years from the date of your sentencing.
While the Pres has the power, it would be the equivalent of catching lightening in a bottle. Nothing in what you have summarized here will advance your cause. Here is the link to the rules: http://www.justice.gov/pardon/pardon_instructions.htm
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One more thing to consider...
ICE (Immigration & Customs Enforcement) apparently hasn't figured out that you were convicted of an aggravated felony. If you file an application for a presidential pardon, the Attorney General will get involved in the background check, and may realize that you are a legal permanent resident (LPR, or green card holder).
As you may know, an aggravated felony is the worst possible type of conviction for a non-citizen. It makes you subject to removal (deportation) and exclusion (being barred from reentering the USA). Even trying to come back into the USA after an aggravated felony conviction is a federal crime that could result in a prison sentence.
In addition to looking at the criminal conviction, you should contact an experienced immigration attorney to see if you're eligible for relief from the immigration consequences. I'm moving this to the immigration category, in the hope you'll get some help on that side.
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I question the decision to post on a forum like this that you did not report the criminal conviction when you applied for the green card. Although you would need a presidential pardon, in my experience an immigration judge is also going to take into account that there seems to be a serious misrepresentation to USCIS as to your criminal history. You need to speak with an immigration attorney with expertise in serious criminal issues.
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The intersection between immigration and criminal law is one of the most complex and convoluted. I think it is imperative that you speak to an immigration attorney before seeking a pardon. Aside from the crime itself, not disclosing the crime on your application is misrepresentation which increases the difficulty of finding a fix for your problem.
The information provided is given for general purposes only. It is not meant to provide any specific guidance or give any legal advice. It is recommended that you bring your specific situation to a licensed attorney for a more complete and thorough evaluation.
Based upon your facts you indicate that you were apparently convicted of the crime prior to becoming a permanent resident and did not disclose your criminal history. While we don't know the facts of your case in detail, you may very well have been eligible for a waiver of inadmissibility. Now in addition to the criminal issue you have a fraud issue as well.
Can you apply for a Presidential pardon? Sure. Anyone can. Is it likely to be granted? As you say, you have a better chance to be hit by lightning. However, before you do anything I would consult with an experienced immigration attorney so that you truly understand the immigration implications of your situation. At the very least, if you are going to request some form of clemency, you want the factual and legal basis of your claim to be correct.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Review Mr. Devore's Avvo Profile for more information about his expertise in immigration law and how to contact him to discuss your case.