In The Entropy Centre, there’s a Trophy/Achievement called Man’s Best Friend, which asks you to “Pet E.” E, in case you somehow missed it, is a charming little Entropy bot who occasionally aids you in your journey. There are two very easy ways to identify E. First, E has a post-it note with the letter E stuck to its forehead. Secondly, E won’t attack you, like other Entropy bots.
You can earn Man’s Best Friend in the very beginning of Act 7. E helps you cross a bridge, after which you can give him a little pat to claim the Trophy. Simply walk up to E and use the button prompt to pet him.
If you want to see a video of this to orient yourself, I’ve got one here:
And that’s how you pet E to earn the Man’s Best Friend Trophy/Achievement in The Entropy Centre.
Also, if you’re looking for solutions to specific puzzles, I’ve got full walkthroughs starting at Act 6, so check those out when you need a little assistance.