The classification of please as an adverb is, I believe, ridiculous.
It can appear in similar positions in sentences (Will you please / kindly / quickly fasten your seat belts - where I believe kindly is not acting as an adverb either – contrast Will you kindly speak to those people who have just seen their dog run over with Will you speak kindly to those people who have just seen their dog run over) - but is being used as a discourse element extra to the semantics of (the rest of the) sentence, whilst usage allows it to be included between the capital letter and the full stop. It is almost the equivalent of an accompanying smile.
This usage of please is, to add to the response number 5 above (sorry, I can't fathom out to whom to accredit it!), a discourse marker subclass politeness (or emphasis if used with irony) marker.
(I've come across this terminology on various websites, and don't know how widely the terms are accepted - or indeed, how widely the courage to sensibly confine the adverb class to words that are truly modifying verbs is to be found.)*
Very kindly would be a two-word variant of kindly; I'd classify it as a single lexeme. Pretty please is similarly a variant of please, but has whimsical or childish overtones, and must occur at the start or end of the request.
- I've been looking for a more sensible approach to word-classes, largely on the Internet, for years now. There still seems to be a lot of argument over where to lump and where to split classes even (or especially) between different linguistics departments. This sadly leaves us saddled with the sacrosanct but untenable 8 classes of antiquity, with people claiming this view as Gospel, with no clear voice suggesting any better alternative.
I've compiled a list as a working model (sorry I can't format better):
Adverbial Particle and other Particle components of multi-word verbs
Conjunction / Coordinator
Prepositional Phrase Modifier
(Sentence connector) . . . (Sentence Adverb) . . . Discourse Marker
Interjection (Nonsense word) (Lyrical filler)
Pro-sentence / Sentence Substitute
and am prepared to consider idioms, open compounds etc as single units (eg He who must not be named; ship of the desert; take off (of a plane); let go; make do...