Example of total order with some properties that is not well ordered

Question

Is there an example of a total order with properties

1. there is a least element and
2. every element has a (unique) successor

not is not also a well ordering?

Answer 1

Yes: $\bigl(\{0\}\times\mathbb N\bigr) \cup \bigl(\{1\}\times \mathbb Z\bigr)$ with the lexicographic order.

Answer 2

If you're unfamiliar with lexicographic ordering, the following set of reals has the same effect (and may be easier to visualize):

$\{1-\frac{1}{n}:n\in\mathbb{N}\}\cup\{1+\frac{1}{n}:n\in\mathbb{N}\}\cup\{3-\frac{1}{n}:n\in\mathbb{N}\}$