"almost whenever" is correct and usable in written English.
You could use it to refer to something that occurs nearly all the time, but not always. For example, "I end up getting take-out almost whenever I don't have time to cook."
In Australia, I have the luxury of requesting the shifts I work and I can take annual leave almost whenever I want.
Vogler's 'calculative view' is accordingly that whenever (or perhaps, almost whenever) you have a reason to do something, you have a calculative reason, that is, a reason whose force is: this is a step toward the termination point of my action.
By which I mean: if I couldn't already switch on an Xbox One almost whenever I like, would this E3 showing make me rush out to buy one?
If you are sleepless or lonely, you can find him on cable, almost whenever, beaming into your brain at maximum voltage, rescuing you from yourself.
In order to enjoy this recipe almost whenever you want, just keep a whole bunch of potatoes pre-sliced in the fridge and all the other ingredients handy as well.
Safeguards can be imposed almost automatically whenever imports create or threaten to create "market disruption".
He putted so well Saturday that it was almost surprising whenever he missed.
But let's be clear: Mr. Lage sounded almost miraculous whenever he was given a clear lane.
Mr. Cameron is himself relatively personable and well spoken, but he tends to look stuffy, almost plodding, whenever he appears next to the verbally pyrotechnic, eternally amused-with-himself Mr. Johnson.
Both accusations emerge out of the law of sexual harassment--a legal area that has come to be invoked almost automatically whenever a case involves sexual intrigue between 2 people at different levels of a chain of workplace command.
The connection is now given by the fact that, in view of [36], every SOLA has a superharmonic representative (that is, they coincide almost everywhere) whenever the measure is nonnegative.
