To "qualify" for an annulment, does not depend on "time"
If you post your state I cana dvise what the criteria is for an annulment. Annulments are rare and expensive, as they require a trial. If you have nochildren in common and no property, your divorce would be a lot less expensive and eaiser to get.

My former fiance got an annullment after being divorced form his wife for six years. It was necessary to get the annullment because they were married in a Catholic Church and she wanted to remarry in the church. It is my understanding that he paid $600.00 to the church to have their marriage anulled. I'd like to hear more on this answer since I never understood the process. Thanks.

Cyn, ( I might be wrong, but i doubt it) your friend received a religious annulment, not a civil annulment.
Annulments are strange legal creatures. They dealwith the concept that "the marriage NEVER happened); an dother equally confusing concepts of "void" and "Voidable"
for instance: here are Nebraska's grounds for an annulment:
Bigamy
If a person is already married and enters into another marriage with someone else without divorcing from the first spouse, the union will be considered bigamous and void. Bigamy is an annulment ground according to Nebraska annulment laws.
Mental Illness
If your spouse is mentally ill and this illness is beyond cure, you can obtain annulment under Nebraska annulment laws.
Consanguinity
Consanguinity is a marital relationship between full-blood or half-blood relatives i.e. marriage with a natural parent, child, grandparent, grandchild, uncle, aunt, nephew, niece, stepparent, stepchild, adoptive parent, adoptive child, brother, half brother sister. Such marriages can be annulled under Nebraska annulment laws.
Underage Marriage
According to Nebraska annulment laws you can obtain annulment, if you are under the legal marriageable age, and have entered into a marriage without the consent of the parents and the court.
Fraud and Duress
Getting the consent by force or duress is as good as committing a felony. If you have been threatened or forced into a marriage, your claim for annulment will be valid under Nebraska annulment laws. hence , the REAL meaning of a shotgun marriage!

Notice there is no ground called "consumating the marriage" nor a "time element"
and , please view Alabama's:
Consanguinity
Consanguinity means blood relations. In Alabama, annulment is lawful if the marriage took place between close relatives like, father, mother, nephew, sister, uncle or aunt. If you have married any such like relative, you can get annulment in Alabama according to Alabama annulment laws.
Existence of Prior Marriage
If your spouse married you when s/he was already married, you can file for annulment in Alabama. You must be single to remarry. Otherwise the marriage will be considered bigamous and it is a valid ground for annulment.
Underage Marriage
You can get Alabama annulment, if you got married before the age of 14. You and your spouse must be more than 14 years to get married, otherwise the marriage will be considered null and void under the Alabama annulment laws.
Under the Influence of Alcohol
If you or your spouse were under the influence of drugs or alcohol when you got married, it won’t be considered a valid marriage. You can get Alabama annulment under this annulment ground.
Impotency
If either of the spouses cannot consummate the marriage, the marriage is considered void. You can file for Alabama annulment according to Alabama annulment laws.
Fraud & Duress
Duress is, when someone forces or threatens you to get married and, fraud is anything done by misrepresentation or by cheating someone. Any marriage which takes place under such like circumstances can be annulled in Alabama under Alabama annulment laws.
and , for further comparison, New York:
Physical Disability
If your spouse has some kind of incurable physical disability that is directly or indirectly affecting your sexual life, you can obtain annulment under New York annulment laws.
Underage Marriage
According to New York annulment laws, both the spouses should be 18 years old before they get married and any younger than this cant legally marry. However such a marriage can be annulled if either of the parties wants to bail out.
Insanity
If your spouse is suffering from insanity for the past 5 years and it is incurable, you can obtain annulment under New York annulment laws.
Duress
A forced marriage is not recognized by the marriage laws of New York. You can file for annulment under New York annulment laws if you have been forced into a marriage.
Fraud
If your consent for the marriage has been obtained by fraud or misrepresentation, you can get New York annulment.

I know there is at least one state that allows for a diffrence of religious grounds But memory fails me)

But , pplease note , none have the common ideas of a short amount of time or failure to consumate ( though impotency is close, but how sexist! what if the female is "impotent, i.e. COLD, would that apply as grounds?

Oh wow. Sorry it took so long to see what you posted. This was in New York on Long Island. I only know what I was told. Thanks for addressing me. Cyn