Is there an example of a total order with properties
- there is a least element and
- every element has a (unique) successor
not is not also a well ordering?
Is there an example of a total order with properties
not is not also a well ordering?
Yes: $\bigl(\{0\}\times\mathbb N\bigr) \cup \bigl(\{1\}\times \mathbb Z\bigr)$ with the lexicographic order.
If you're unfamiliar with lexicographic ordering, the following set of reals has the same effect (and may be easier to visualize):
$\{1-\frac{1}{n}:n\in\mathbb{N}\}\cup\{1+\frac{1}{n}:n\in\mathbb{N}\}\cup\{3-\frac{1}{n}:n\in\mathbb{N}\}$