Question: Press the Attack is a new rune during this preseason's rune changes. It's effect causes enemies to take 12% more damage from all sources if you basic attack them 3 times. 

Some champions have abilities that enhance their auto attack damage. For example, Vayne's passive does extra true damage if she attacks the same target three times. Ezreal's Q applies on-hit effects, so it's (kind of) a skill shot basic attack. Nasus' Q does more damage the more stacks he has. Etc. etc.

Does the extra damage from these "enhancements" get amplified by 12% on the third attack, or does that boost only start after Press the Attack has been triggered (fourth attack and onward)?

Answer: 
**As of patch 7.24 all bonuses apply on and after the 4th hit**

So I just tested this in the practice tool and the 12% bonus damage are applied on the 3rd hit for both the **initial attack and the bonus effect**.

The 12% damage taken increase will be applied **before** the damage.

For testing I used Siphoning strike (about 1.4k stacks) and Sheen on Nasus and I've also tested Kayle's E. Other attacks and effects should behave the same way.
An update to Jutschge's answer:

Since [patch 7.24](https://na.leagueoflegends.com/en/news/game-updates/patch/patch-724-notes), Press the Attack no longer applies the damage amplification to the 3rd hit, but rather after the 3rd hit.