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