The random variable $T_i$, the "Hitting Time of $i$" is defined to be the first $n$ such that $X_n=i$ given that $X_0=i$.
By the mean recurrence time of $T_i$, I mean the expected value of this random variable.
I wish to show that if $i$ is transient, then the expectation does not converge to any finite real number. While this, intuitively makes sense, I do not know how to formally prove this and any help is appreciated.
On
Note that state $i$ is persistent iff, $$ P(X_n = i \text{ for some } n \geq 1| X_0 = i) = 1$$
Each state is transient or persistent.
The hitting time of state $i$, $T_i$, is a random variable defined as the first time we visit state $i$:
$$ T_i = \min \{n | n \geq 1, X_n = i\} $$
where $T_i$ is defined as $\infty$ if this visit never happens.
We now show that $ P(T_i = \infty | X_0 = i) > 0 $ iff state $i$ is transient. Then, the required result on the mean recurrence time follows, because the mean recurrence time $\mu_i$ is defined as: $$\mu_i = E(T_i| X_0 = i) $$
Suppose state $i$ is transient. Then, $$ \begin{align} P(T_i = \infty | X_0 = i) & = P(X_n \neq i \text{ for all } n \geq 1 | X_0 = i) \\ & = 1 - P(X_n = i \text{ for some } n \geq 1 | X_0 = i) \\ & > 1 - 1 = 0. \end{align} $$
Suppose state $i$ is persistent. Then, $$ \begin{align} P(T_i = \infty | X_0 = i) & = P(X_n \neq i \text{ for all } n \geq 1 | X_0 = i) \\ & = 1 - P(X_n = i \text{ for some } n \geq 1 | X_0 = i) \\ & = 1 - 1 = 0. \end{align} $$
This shows both directions, and completes the proof.
I don't think you have the definition quite right. The first passage time $T_i$ is the minimum $n$ such that $X_n = i$ given $X_0 = i$, and is defined to be $\infty$ if no such $n$ exists.
In that light, we just need to know that a state $i$ is transient if there's some other state $j$ such that $i \rightarrow j$, but $j \not\rightarrow i$. That is, if a (finite-state, discrete-time) Markov chain initialized at state $i$ reaches state $j$ it almost surely never returns to $i$. Because $i \rightarrow j$, this happens with positive probability and thus $T_i = \infty$ with positive probability. (Which implies $\mathbb{E}[T_i] = \infty$).
One reference you may find useful is these lecture notes.