"Do you know the story of the 7 hour war?"
This little bedtime story doesn't end well for the Human race. You already know things aren't good by the Orwellian surroundings of City 17, but by scouring Eli Vance's lab outside the city, you can find newspapers reporting on the fate of humanity.
The papers cry out "Earth Surrenders", and "Portal Storm Continues", also explaining that Dr Breen was declared 'Interim Administrator'. And, of course, that the "7 Hour War Ends in Combine Victory". Unfortunately, that's all that is readable. But it's very easy to miss.
"The Blue Screen of Death returns!"
Dr Kleiner suffers the same problem that hit Black Mesa - just peek at his computer screens in the lab Alyx takes you to at the beginning of the game. One monitor displays this famous Microsoft error. So that's why you end up teleporting all over the shop.
"The cheats are a reference to Quake"
Well, one cheat in particular. Back in the day, Quake on PC needed a simple command input in the 'console' to unlock different items, weapons, etc. The command was 'impulse 11" to get all weapons. In H-L2, it's "impulse 101" for the same result. It doesn't get much geekier than this.
"Lamarr loves watermelons, Dr Kleiner was a cover star"
Two more easily missed titbits here, in Kleiner's City 17 teleporter lab. Stuck to the border of his notice board is a post-it note that reads "find more watermelon for Lamarr". And just above this on the board itself is a fetching picture of a young Kleiner - complete with hair - adorning the cover of Popular Science magazine.
"Have you met the wise, but bonkers Vortigant?"
Hidden away in the canals is a Vortigant with many a phrase to utter. After completing the Water Hazard chapter, and defeating the fourth helicopter, you'll end up in a huge area with lots of dead end routes and a ramp. Stop, and look out for two circular pipes with grates over the ends.
Park the boat under the grate that's slightly open and leap in. It'll take some doing, but it is possible. Once inside, continue along the pipe and use the HEV boost to zip over the radioactive waste and up onto the far ledge. Here you'll find a Vortigant cooking spitted headcrabs over a fire and chanting to himself. Press the 'use' button on him to recieve his mighty wisdom.