Can refugee apply to adjust status after one year has passed?
It's legally possible to apply late, but safest to include an explanation of the delay.
It’s been two years since I came to the U.S. as a refugee, I have not applied for a green card, and I hear I was supposed to do so after one year – can I still apply, or have I missed my chance?
Yes, you can still apply. It is true that the law says that, as a refugee, you should have applied for the green card as soon as you became eligible for it, which was at the end of your first year in the United States.
But the law does not necessarily say that you cannot apply for the green card after two years or more. Still, we should keep in mind that U.S. Citizenship and Immigration Services or USCIS (the main government agency responsible for approving or denying green card applications) has a lot of power (or "discretion") to deny your green card application if the agency feels that you do not deserve it. In other words, it is not impossible that a USCIS officer would feel that you, as a refugee, do not deserve a green card simply because you violated the law by waiting too long before applying for the green card.
Since the law in this regard is not very clear, and it is unclear whether USCIS really has the authority to deny refugee green card applications solely on the ground that they were filed late, immigration attorneys often recommend that refugees add to their green card applications a written explanation of why they applied late, as well as a description of positive things they are doing (or have done) in their new life in the U.S. (for example: steady employment, family and community involvement, and so on).
Another thing to consider is that, although Immigration and Customs Enforcement or ICE (the government agency responsible for enforcing the immigration laws) does not currently consider your failure to apply, in itself, a sufficient reason to detain you or remove you from the U.S., nonetheless, not applying for adjustment of status makes you more vulnerable to arrest and detention on other grounds, especially if you commit a crime. This is one more reason why immigration attorneys recommend that refugees apply for adjustment of status regardless of how late their application is.