Three years ago, I was deported from the U.S. because I was caught after overstaying my visa. After I returned to my country, I became involved in politics, and received death threats. So I had to flee. I came back to the U.S. last year (by crossing the Mexico border without being seen). Now I want to apply for asylum but I am afraid I won't get the chance, because of having been deported. Can I still apply for asylum?
You could certainly try applying for asylum. However, whether your application will be accepted (let alone approved) is a difficult question, because this is not a very clear area of U.S. law.
On one hand, the part of the law that defines how people can become eligible for asylum does not explicitly bar applications by people who illegally reentered the country after having been ordered removed. On the other hand, another part of the law explicitly says that noncitizens who reenter the U.S. after having been removed may not apply for any form of relief—which would presumably include asylum.
Up to now, the U.S. government has interpreted this bar narrowly enough to comply with international laws that prohibit returning people to countries where their life or freedom might be in danger, or where they might be tortured.
Applicants were at the very least be able to obtain a "reasonable fear" interview with the Department of Homeland Security (DHS) to find out whether they might qualify for withholding of removal or protection under the United Nations Convention Against Torture. (Be aware, however, that going forward with this request so would probably entail exposing yourself to immediate detention and, perhaps, to summary removal—without a chance to defend yourself in immigration court.)
Under the Trump Administration, however, rules proposed in late 2019 would bar asylum to applicants who had committed an offense under 8 U.S.C. § 1326, namely illegal reentry.
An alternative would be to try to undo that removal order by reopening your previous immigration court proceedings. Since the events that led you to flee your country took place after you were removed, you could in theory argue that your prior removal proceedings should be reopened based on changed country conditions. Be aware, however, that immigration judges would have considerable discretion to grant or deny your request (your motion to reopen).
In sum, it is unclear whether or not you would be able to apply for asylum, and depends on whether the 2019 proposed rules go through (as they are likely to). But, at least, the arguments at your disposal do not seem "frivolous" with regard to applying for asylum.