Is machine epsilon the largest relative error in representing a number as a floating point number?
In the most common usage, and in particular in IEEE-754, machine epsilon is the smallest number $\epsilon$ such that $1 + \epsilon \ne 1$.
It does have an interpretation in terms of relative distance between numbers: It is the largest relative separation of two consecutive normal numbers.
There are a couple of important points to understand here:

This is only about normal numbers, as opposed to subnormal numbers (i.e. what older standards called &quot;denormal numbers&quot;).
It is about consecutive numbers, namely, some floating point number and its closest neighbour.
Machine epsilon measures separation, not error. It is the relative distance between those floating-point numbers, not the relative error between a floating point number and anything.

So if $x_1$ and $x_2$ are consecutive normal numbers, then:
$$0.5\epsilon \le \frac{|x_2 - x_1|}{\max(|x_1|,|x_2|)} \le \epsilon$$
So if you have a candidate real number, and you want to represent it as a floating-point number, how bad can the relative error be?
As gnasher729 rightly pointed out, it can be as large as you want if the real number would overflow, underflow, or be subnormal. But assuming that it's in a reasonable range, meaning that it is between two consecutive normal numbers, then one of those normal numbers is within $0.5\epsilon$ of it in a relative sense.
Because this is a fraught definition, the term that we usually use is &quot;units of last place&quot;, or &quot;ulp&quot; for short. If a real number is within the normal number range, there is a floating point number whose error is at most 0.5ulp.