Ah so just off by 1 :)

Now I remember, the "canonical" transcendental number is sum of reciprocals of n! Which is almost rational.

Precisely. In fact, that features in many proofs of transcendence: you find too good rational approximations, which contradicts things like Roth's theorem, so your number cannot be algebraic.

IIRC the first number proven to be transcendental was Liouville's number

0.110001000000000000000001...

where the digits after the decimal place are 1 in the n!th place and 0 otherwise. This is explicitly constructed to be very close to the sequence of rational numbers

0.1, 0.11, 0.110001, ...

(I expect there's nothing special about base 10 here; surely the proof works in binary as well.)

So actually the number you're referring to the natural base of logarithm, usually written "e".

It turns out that e has the same irrationality measure as irrational algebraic numbers (2), meaning it can be approximated similarly well as irrational algebraic numbers, and not as well as some other transcendstal numbers like Liouville's constant

Actually integers are very poorly approximable by rational numbers, even though each integer is very well approximated by a single specific rational number

The definition of "well approximated" is that there are infinitely many good approximations, not just one really good one, and this is what integers, and algebraic numbers in general, fail to have

Every integer has an infinite number of combinations of integers that add up to it.

Indeed!

Mathematics is a lot about defining something and then proving stuff (and sometimes going the other way around). Different combinations giving the same approximation are considered a single approximation, namely the result of the combination