When doesn't a supremum exist? Other than ∞, is there another case where a supremum (or an infimum for that matter) doesn't exist?

Within the extended line $[-\infty,+\infty] = \mathbb R\cup \{\pm\infty\}$ every subset has a supremum and an infimum.  Within the line $(-\infty,+\infty) = \mathbb R$ every subset has a supremum and and infimum except when the supremum or infimum within the extended line is $-\infty$ or $+\infty$.  For example
$$
\sup \{1,2,3,\ldots\} = +\infty
$$
and
$$
\sup\varnothing = -\infty.
$$
