I What is one (1)?

At bottom, this is a philosophical question. There are differing theories about the ontology of mathematical objects. Some schools of thought are:
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Platonism
Nominalism
Conceptualism
Fictionalism
Meinongianism

I have provided links to the first two. If that doesn't keep you busy, feel free to search for the others.

 

One is an adjective. An adjective gives information about a noun. Quantities of any named noun serve the purpose of "how much" or "how many". For the rest of your answer, look again at post #2, of jedishrfu.

 

You might want to look at abstract algebra. Considering the integers as a group with the operation of addition, '1' generates the entire set with repeated addition (and subtraction). When the operation of multiplication is added to the picture, '1' is the identity ( 1 * x = x ). In abstract algebra, the basic properties of '1' are defined and studied.

 

One is the loneliest number that can ever be. Two can be as bad as one...

Seriously though. I find the mind at 3:00 am often finds profundity where none exists. My brother and I were cutting through the philosophy building on campus and we came across a door with a broken glass window (ostensibly, to see if someone was on the orher side of the door, before opening it too violently) The glass had a sign on it, "broken glass" My brother suggested the sign was put up by a maintenance person to avoid arguments. Were it not for the sign, the philosophers in the department would argue, "maybe the glass is all fixed, and they are the ones that are broken."

BTW, I am told you cannot count to much over 5 in a dream. Your (any) mind in that state does not allow that structure.

 

Apples can be continuous when transformed into applesauce.

 

''one'' is only a word used to denote the symbol ##1##. Humanity uses the symbol ##1## in order to ''count things''. From ##1## you can generate other numbers by the ''successor'' ##S(1)=2##, it is the minimum of the set ##\mathbb{N}## (natural numbers without zero). It is the neutral element of the multiplication, this answer partially to the first question:

For the second, how we can measure ##1##? I measure physical quantities not numbers as ##1## apple, ##1 L## of water ... In a certain sense I can count the number of times that in this thread I used the symbol ##1##.
Measuring one is like to say ''I have ##1## one'', this is ambiguous from a rigorous point of view ...

 

Despite numerous answers that imply that there is no single meaning, you continue to search for one. Perhaps you should ask yourself why you are putting so much emphasis on one of an uncountably infinite number of numbers. While it is surely true that in studying various meanings of 1, your understanding will grow, that is equally true of anything else you study. Numbers are abstract concepts which occupy no space, no time, and have no mass. Although it can also be argued that concepts occupy space-time and organization (structure, rules) in the minds of the beholders. You really, really need to NOT introduce other concepts without first defining them (e.g. dimension). Its well known what happened when Russell & Whitehead tried to axiomatize mathematics - it's unlikely to provide fertile ground 100 years later. So, rather than chase your tail or reinvent the wheel, try to get a grip on why going over this well plowed ground is going to be useful to you. My personal take is that the Universe is the one true thing, and any division of it into separate objects must be imperfect and incomplete. Think about this: if it wasn't useful, would you still be interested? If it is its utility, then consider whether "what" is more important than "how". (Leaving "why" to the philosophers)

 

Wikipedia: "In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is thus a bare number, and is therefore also known as a quantity of dimension oneˇ"

Understand it as a quantity of dimension [one]. One is a dimension like L (length) or M (mass) and so on. So 1 does not have dimension, but it is dimension.
It is also the unit, the smallest nonzero measure of a countable quantity. You can "measure" (count) the number of apples in a basket, and you get 1 when there is only one apple, so you can get 1 as measurement result. When there are two apples, the number of apples is two ones :)