This will depend on what statistics you use to determine statistical significance.
If you are using something like a Ttest, you really want as many replicates as possible. You can't do these tests without at least 3 replicates. Depth isn't particularly useful here, but more replicates are. It's an oversimplified approach to the statistics, IMO.
If you use something like Fisher's Exact Test (hypergeometric or poisson distribution) then two biological replicates should be reasonable. You can actually mix multiple biological replicates in the same barcoded sample and it will likely give the same answer (since the reads from replicates are just added together). In this case the read depth is more useful than additional replicates. This method is what we use, but a number of people (probably more knowledgeable than me) have raised concerns with it.
If you use something like DESeq I don't have an answer for you, because I don't know anything about their statistical models. My guess is that two biological replicates would be fine for this type of analysis, though you may need 3.
I think this is a useful question, though, and I am interested to see what others think.
