How exactly does the Endless Bonus mechanic work?

Solution 1:

From the wiki:

Every time you open a Relic, you get a predetermined booster. These boosters stack the longer you stay.

• The first interval: 1.25x Affinity.
• The second interval: 1.25x Credits.
• The third interval: 1.25x Resources.
• The fourth interval: 1.25x Resource Drop Rate.
• The fifth interval: a random Exceptional Relic (that can be used in the same mission).
• The sixth interval: 1.5x Affinity (replaces first).
• The seventh interval: 1.5x Credits (replaces second).
• The eighth interval: 1.5x Resources (replaces third).
• The ninth interval: 1.5x Resource Drop Rate (replaces fourth).
• The tenth interval: a higher tier random Flawless Relic (that can be used in the same mission).
• Boosters continue to increase every set of intervals and are capped at double (2x). After the nineteenth interval, all boosters reached their cap.

After the first "round", you get the first interval bonus. Same goes for all other bonuses. The relic that drops is directly sent to your inventory, allowing you to immediately use said relic during the next round's relic pick. It is not an end-of-mission reward, as the wiki clearly states that it can be used in the same mission, and it is for that reason you will not see it in the mission reward list.