Quasipoisson regression goes beyond standard poisson regression in taking into account overdispersion (whereby the dependent variable's variance is much greater than its mean). This is explained at this article.

But how much dispersion is so much that it should rule out even the use of quasipoisson regression?

In one highly upvoted answer on this site, quasipoisson is used even when the variance is 206 times the mean. Is this sound?