Drawing samples conditioned to be larger than the previous, until a threshold is hit Suppose I have some probability density function $f$ and a threshold $t$.
Let $X_0 = 0$
I now recursively draw $X_{i + 1}$ using the distribution defined by $f(x | x \gt X_i)$ until $X_k \gt t$ for some $k$.
What's the name for this sort of process?  What can I say about $\sum_1^k g(X_i)$ for some function $g$?
 A: As whuber points out in the comments, your problem can be considered for the uniform distribution without loss of generality, since every continuous random variable is a monotonitc transform of a continuous uniform random variable.  Since a uniform random variable with an imposed condition of a lower-bound is still uniform (over its new conditional range) the problem further simplifies into a problem that can be written as a transformation of an underlying IID sequence of uniform random variables.
It turns out that this problem leads you to some analytically complicated results, relating to the use of the inclusion-exclusion summation form on a set of uniform random variables.  I cannot get a full analytic solution to your problem, but here is substantial progress towards a solution.

Specifying the problem: Let $U_1, U_2, U_3, ... \sim \text{IID U}(0,1)$ and define the corresponding series of interest $X_0, X_1, X_2, X_3, ...$ by the recursive equations:
$$X_0 = 0 \quad \quad \quad \quad X_{t+1} = X_t + (1-X_t) U_{t+1}.$$
It is easily verified that $X_{t+1} | x_t \sim \text{U}(x_t,1)$, which is the desired conditional distribution in your problem.  It is also notable that the sequence of values of interest is strictly increasing (which is a property we will use in the solution below).  Now, for any argument value $0<t<1$ we define:
$$K(t) \equiv \min \{ n \in \mathbb{N} | X_n > t \}.$$
This is the index for the first value that exceeds the argument threshold.  To solve your problem we need to find the distribution of $K(t)$.  This then allows you to examine the sum you are interested in by applying the law-of-total-probability.

Finding the distribution of $K$: To find the distribution of interest, we first note that the values of the series of interest follow the exclusion-inclusion recursive form and so they can be written non-recursively as:
$$X_t = \sum_{\mathcal{A} \subseteq \{ 1, ..., t \}} (-1)^{|\mathcal{A}|-1} \prod_{j \in \mathcal{A}} U_j .$$
(The reader is invited to verify that this form satisfies the recursive form specified above.)  Since $X_0 < X_1 < X_2 < X_3 < \cdots$ (i.e., the sequence is strictly increasing) we have:
$$F_K(k) = 1-\mathbb{P}(K(t) > k) = 1-\mathbb{P}(X_k \leqslant t) = 1-\mathbb{P}\Bigg( \sum_{\mathcal{A} \subseteq \{ 1, ..., k \}} (-1)^{|\mathcal{A}|-1} \prod_{j \in \mathcal{A}} U_j \leqslant t \Bigg).$$
This probability is analytically complex, but it is relatively simple to estimate by simulation using $k$ uniform random variables.  You can do this by direct specification of $X_k$ via its non-recursive form.  I am not aware of any known analytical form for this distribution, but perhaps others on this site can supplement this analysis with further insights.  Obviously, once you have the distribution of $K$, you can then obtain some insights into the sum of functions of the first $K$ terms.

