What "undocumented" benefits are there to being an "elite" frequent flyer?

My father used to be one, and that was pretty much it. Priority for upgrades, being taken more serious when you have complaints or requests, and that's about it.
And not getting a form letter from the CEOs secretary but a personal letter written by the man himself when he complained about lacking maintenance he'd noticed developing over a period of several months (broken cabin equipment not being repaired, even after repeated complaints, 2 weeks after writing the CEO he flew the same aircraft again and it had been fixed when for months prior it had been broken, probably no coincidence).
Flight attendants knowing you by name and remembering your drinks preferences are normal in business class with quality airlines for frequent travelers on a route, has less to do with being a high ranking card member than with them simply serving you several times a week over a prolonged period.

You get invitations to PR events hosted by the airlines such as introductions to new business-class layouts, occasional concerts etc.

Most benefits are public, which provides incentives for people to attain a higher elite level. As you suggest, there are benefits which are not public. These fall into two categories: official non-published benefits are based on internal airline policy, available to customer-facing employees in the relevant work areas (such as reservations, flight attendants, etc.) Unofficial benefits are casual benefits extended by an individual airline employee, as a matter of custom or as a favor, but are not based on airline policy.

Airlines may establish official unpublished policies in an effort to simultaneously (a) retain high-value passengers who might otherwise defect, and (b) keep the benefit quiet to avoid having large numbers of people request it, or to retain the flexibility of changing the policy quietly, or simply to keep the published benefits simpler.

Individual airline employees may extend unofficial benefits out of a sense of fairness or in an effort to keep high-value customers happy.

It's difficult to get into details without being limited to a specific airline program, but examples of official non-published benefits include:

• Free food and/or drinks when otherwise not entitled (such as flying in domestic coach or in a domestic airline lounge).
• "Hidden" award inventory (award seats only available to members at a high or top elite level)
• Ability to use system-wide upgrade instruments for flights later than the expiration date
• Earning additional system-wide upgrade instruments or redeemable frequent-flyer miles for travel beyond a threshold
• Making desired inventory available on special request (such as opening an award seat or a discount fare bucket)

Examples of unofficial benefits include:

• Upgrades on non-upgradeable tickets (e.g., an award ticket in certain programs, discount inventory in certain programs)
• "Protection" during irregular operations (being booked into alternate flights as a backup while still holding confirmed space on original flights when the original flights are delayed)
• Being admitted into a lounge not permitted by status or class of service (e.g., a first-class lounge when flying domestic coach)
• Changing flights without fee (when a fee would normally apply)
• Receiving a larger number of bonus miles as an apology when something went wrong

If you are an elite flyer (by elite I mean the real elite, not just the top of their frequent flier ranking) you can get some ridiculous perks.

Emirates, for example, has the iO (Invitation Only) tier. As in the name, you are selected and invited to this tier (by Tim Clark - president of Emirates) - and your invitation lasts for two years, upon which it may be renewed depending on your travel history.

It is not clear how one can get this invitation however rumors are that if you book more than 50 full fare business class fares per year, you can get invited.

Some of the known perks are:

1. Card individually delivered to you by hand by a senior Emirates sales manager.
2. Matching luggage tags, Mont Blanc wallet and a complimentary Gold membership card for a companion.
3. Just like Ryan Bingham from Up in the Air each card has a private direct line.
4. Guaranteed Business Class seat on any flight.
5. Limo rides to/from the airport, to/from the airplane no matter what class you book (even Economy).
6. Access to whatever is the highest class lounge available for Emirates or Qantas - this means, First Class for Emirates and Qantas Chairman's Lounge
7. If you are traveling business, and there is room in first, you'll be moved there before landing so you can make a quick departure.
8. You can get very specific catering requests; and they are ensured to be delivered to you.
9. Free invitation to the sporting events (of which Emirates is a sponsor).

... and some more.