The set \(\displaystyle S\) of rational numbers \(\displaystyle x\) with \(\displaystyle x^2<2\) has rational upper bounds but no least rational upper bound. Are there any sequences that have upper bounds but no least upper bound (or have lower bounds but no greatest lower bound)? Thank you.

The question is ambiguous since you haven't defined what your "universe" of numbers should be. If you work in $\mathbb{Q}$, then the example you gave is exactly the thing you are asking for. You are just stating it a different way. The set $S$ you described clearly has an upper bound. However, it has no least upper bound.

On the other hand, if you work in $\mathbb{R}$, then this can never happen since $\mathbb{R}$ is complete. In fact, this is one way of defining the reals.