Q:

What is a good prototypical example of a construction that is not well-defined?

In the question Why do students have problems with showing that something is well-defined? How can this be improved?, it was suggested that perhaps students have never seen something that is not well-defined.
I agree with this and would like to hear what you think is a good starting example for a not-well-defined object. I think a good starting example should require as little background knowledge as possible while avoiding the pitfalls of the following examples:

Too fixable: You could claim that "the antiderivative of a function" is not well-defined, since there are lots of them. But students will easily protest that the antiderivative can just be something with a +C on it, which seems to clear up the issue. This doesn't help anyone who doesn't already understand the idea.
Too obvious: You could say that "the denominator of a rational function" is not well-defined (or similar things in the link above), but the students see it immediately. This gives the wrong impression; in fact checking well-definedness carefully is critical and you can almost never see it immediately.

A:

Supposedly defining a ring homomorphism $f:\mathbb Z/m \to \mathbb Z/n$ by $f(k)=k$ is a popular, traditional attempt to define a map ... initially $\mathbb Z\to \mathbb Z/n$, that may be imagined to "factor through" the quotient $\mathbb Z/m$, but which does not without assumptions on $m,n$. Yet, in my experience, students look at the "formula" for it, and cannot see the difficulty, since the formula has such a straightforward appearance. :)

A:

The naive "addition" on rational numbers, denote it $\oplus$, where you simply add numerator with numerator and denominator with denominator.
Example:
$\tfrac 1 3 \oplus \tfrac 3 4 = \tfrac 4 7$.
I think this is less obviously non-well-defined than the other examples involving fractions.

A:

The linked question has a decent example, from mweiss: 

"Well-defined" only becomes a meaningful concept if you have
  experience with cases in which something is not well-defined. Here is
  a simple case in which something seems entirely reasonable: Let $m,n$
  be two integers and $[m],[n]$ their equivalence classes mod $p$.
  Define $[m]^{[n]}=[m^n]$. Seems reasonable, especially because of the
  way we define the other arithmetic operations mod $p$. But as soon as
  you try to calculate particular examples you realize the definition is
  broken; different representatives of $[n]$ yield different results.

