Learn English – Adverb “already”

Note that there are a few instances of "been being carefully" in Google Books, but no relevant occurrences of those three words in any other sequence.

Personally I have no real preference for carefully being placed before or after the verb - but in this context, "the verb" means the word "tickled", not the "helper" verb forms "been" and "being". Thus,

I have been being carefully tickled.
I have been being tickled carefully.

...are equally valid, and mean exactly the same thing. Here are a couple of real-world instances...

...has been being researched systematically...
...have been being carefully researched...

In short, put your adverb right next to (either immediately before or after) the "operative verb". Just don't splice it in between any pair of "auxiliary" verbs you happen to be using.

Adverb placement is not fixed, and as with other elements, I think most of it depends on what you want to express.

"He has always been an academic and a charitable person". This is a kind assessment of a man, and also the simplest and most common construction.

"He always has been an academic and a charitable person.* This is more emphatic, as if in defense of some alternate version of his history.

"He has been a wise and, always, a charitable person." This emphasizes the constancy of the latter attribute. All are "correct".

"We had been slowly drifting down the river when a bear attacked." is no different really than "We had been drifting slowly down the river when a bear attacked." To my ear (AmE), the latter sounds a bit more common. I agree it would be unusual to break up had been in this case, but "he had often been seen..." is not an unusual construction.

"He had been either sick or exhausted." is much the same as "He had either been sick or exhausted." Here, either is emphasized because of it's less common placement. Both are grammatically correct.