Here is perhaps the simplest example. Let $A$ be the C*-algebra of all sequences of $2 \times 2$ matrices converging to a scalar multiple of diag(1,0). Let $p$ be the constant sequence diag(1,0), and q a sequence of rank 1 projections converging to diag(1,0) but never exactly equal. Then $p$ and $q$ have no supremum at all. This example can be tweaked to make it unital by allowing any limit matrix at infinity and taking $q$ to alternate diag(1,0) and nearby but unequal projections. Then $p$ and $q$ have no least upper bound.

Here is perhaps the simplest example. Let $A$ be the C*-algebra of all sequences of $2 \times 2$ matrices converging to a scalar multiple of diag(1,0). Let $p$ be the constant sequence diag(1,0), and $q$ a sequence of rank 1 projections converging to diag(1,0) but never exactly equal. Then $p$ and $q$ have no upper bound at all. This example can be tweaked to make it unital by allowing any limit matrix at infinity and taking $q$ to alternate diag(1,0) and nearby but unequal projections. Then $p$ and $q$ have no least upper bound.