Consider a Markov chain in a finite space and with two absorbing states, each of which is accessible from the other, transient states. Is the expected number of transitions to reach any single absorbing state infinite?
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Consider a chain with 3 states: $1,2,3$, the first and the last being absorbing and $2$ having transitions of $\frac12$ probability to each of the latter states. Then the expected times are $$ \begin{align} (0,1,\infty) &\text{ for reaching state }1 \\ (\infty,1,0) &\text{ for reaching state }3 \end{align} $$ Clearly, the expected hitting times from one absorbing set to another is infinite, but for all transient states it easily can be bounded.
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