The AADSAS application accepts more than three letters! However, please be aware that admissions faculty is only required to read three letters, regardless of how many you submit. For example, imagine a scenario in which you submitted five letters, two of them were strong letters and three of them were mediocre. If three of the letters were read at random the likelihood of getting the three letters you wanted read are greatly reduced. The following table shows the probability of each combination:
|2 Good/1 Mediocre
|1 Good/2 Mediocre
|0 Good/3 Mediocre
Based on the assumption that an admissions officer will only read three letters, 1/10 schools you applied to did not get the opportunity to read any of your top letters, while 6/10 schools only got to read 1 of them. As you can see, the odds are not in your favor. For this reason, I highly recommend following a quality over quantity ideology when submitting letters of recommendation. Make sure to get strong letters and pick the letters that you believe will carry the most weight. It will always be better to have three very strong letters rather than 5 moderately strong letters.