Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is
$$\Phi(n)=n-\sum_i\frac n{p_i}+\sum_{i \lt j}\frac n{p_ip_j}-\cdots+(-1)^{k}\frac n{n}=nr$$ where $r=\prod(1-\frac 1{p_i})$.
For any positive integer $x$, the number of integers less than or equal to $x$ and relatively prime to $n$ is given exactly by the alternating sum $$\Phi(n,x)=x-\sum_i\lfloor{x/{p_i}}\rfloor+\sum_{i \lt j}\lfloor x/{p_ip_j}\rfloor-\cdots+(-1)^{k}\lfloor x/{n}\rfloor$$
How large and how small can the error $\Phi(n,x)-xr$ be? Can the absolute value of the error exceed $k$?
Certainly the error can be no more (in absolute value) than $2^k$, probably it is easy to show that it could not be more than the middle binomial coefficient $\binom k{\lfloor k/2 \rfloor}$. I can see that it could get (almost) as large as $ k $ by imitating the following example:
The integers $1122659, 2245319, 4490639, 8981279, 17962559, 35925119, 71850239$ are a Cunningham chain as is $2,5,11,23,47$ in that each member is prime and one more than twice the previous one. If $n$ is the product of the $7$ primes from the first chain and $x=1122659\cdot64-1=71850175$ then all seven terms $ x/{p_i} $ are just a bit less than an integer so, without the rounding down to an integer, the estimate $rx$ will too small by about $7$ (the other terms are quite small). Of course it is not known for sure that there are arbitrary length chains. Maybe a similar idea could get an error of order $2k$ or $k^2.$
later Thanks for the answers. I give one of my own below explaining that actually the best we could hope for is $2^{k-1}$ and then exhibiting a construction of D. H. Lehmer which attains $(1-2/q)2^{k-1}$ for arbitrary $q$. This is pretty much a result of the other answers, but I thought it was worth showing off the construction (which is not immediately clear from the article).
Best Answer
First of all as you remark above, it is easy to see by elementary means that the error is at most $2^k$. It was an old conjecture of Erdos that this error term could be improved to $o(2^k)$. However this conjecture turns out not to be true!
In 1951 Vijayaraghavan proved that the error term $O(2^k)$ is best possible in the sense that for any $k\in\mathbb{N}$ and $\delta >0$ you can find an integer $n$ with exactly $k$ distinct prime factors, and a real number $x$, such that
$$\Phi(n,x)-xr>2^{k-1}-\delta.$$
Vijayaraghavan's paper is On a problem in elementary number theory. J. Indian Math. Soc. 15, (1951).
For a more detailed and up to date discussion you could look at the more recent paper Codecà, Nair, An extension of a result of Lehmer on numbers coprime to n. Ramanujan J. 16 (2008), no. 1, 59–71.