Example of a general random variable with finite mean but infinite variance
Solution 1:
One answer is Pareto distribution with parameters $\alpha , x_0$ which are both positive. The distribution is given by:
$$f_X(x)= \begin{cases} \alpha\,\frac{x_0^\alpha}{x^{\alpha+1}} & x \ge x_0, \\ 0 & x < x_0. \end{cases}$$
Note that $E[X] = \infty$ for $\alpha \leq 1$ and is finite elsewhere.
The variance is not finite for $\alpha \in [1,2) $
Hence it satisfies your question for $(1,2)$
In general,$E[X^n]= \infty \ \ ;n\geq \alpha$.
EDIT: Clarified the answer as suggested.
Solution 2:
An example is a random variable $X$ having a student-t distribution with $\nu = 2$ degrees of freedom
Its mean is $E[X] = 0$ for $\nu > 1$, but its second moment $E[X^2] = Var[X] = \infty$ for $1 < \nu \le 2$
Edit: Oh wait, finite positive. Well, $X+1$, I guess.