Sure - we've been doing these types of negotiations for senior and director-level (even CTO-level) engineers for over 5 years now. We have helped engineers negotiate employment packages with Facebook, Google, Stripe, HSBC, Apple, Salesforce, Instacart, FireEye, Attune, Palo Alto Networks, and many more.
While I don't have a hard statistic/number for you (we are working on putting those numbers together now) our most recent client got over a 100% increase over the initial offer. We also had a client that got so much more than they were initially offered that their future boss also got a raise because the company had to rethink the pay structure of the entire division.
I understand if it's hard to calculate the median. But what's the minimum increase?
Some of the companies you mention try hard to make the initial offer compelling/fair (e.g. slightly above max of {current compensation, competing offer, target compensation for level}), so I'd assume that sometimes they simply won't budge beyond that, as they feel that they've taken into account all relevant info before making the offer.