Power analysis for unknown experiment duration How would I do a power analysis if I do not know how long we want to run the study?
I want to use analysis to inform how long I want to run the experiment based on how long it will take to gather the samples needed. The part that I do not understand is - if I do not know how long we want to run the experiment, how do I come up with the mean and standard deviation needed to plug into power calculation since mean and sd will change over time? For example, if I want to measure sales per user, multiple sales can happen over time.

This is an important issue for metrics that accumulate over time, like sessions or revenue per user.
You need to use your historical data to see how the number of unique users N, the mean m1 of the user-level outcome, and the standard deviation sd of the user-level outcome increase with the duration of the test. You update these by calculating the running version of these three by day. You will also need to adjust the data if there are strong trends or seasonality around the time of the test. You don't want to assume that you get your peak traffic in designing the test if conditions during the test will be worse.
Once you have your three columns, you do the usual minimum reliably-detectable effect (MRDE) calculation where N, m1, and sd move in tandem rather than sending N to infinity while keeping m1 and sd fixed.
This returns something like this with a simulated dataset but real MRDE calculations for a two-sided t-test:

Here you get ~5K users on day 1, with a mean of 17.27 and a standard deviation of 19.26. Splitting them into equal experiment groups and assuming sd1=sd2=sd, you can detect a delta of 1.51 (in levels) or 9%.
On day 2, an additional 15K users show up for a total of 20K. The updated mean is now higher, and so is the sd. The delta is now 1.73, but in percent, that is 6%, so that's an improvement.
On day 3, you only get 7.7K new users, so the MRDE does not drop that much.
You keep going until you find an MRDE that you can live with. That pins down the approximate test duration. To detect a 4% lift, you need at least a 17-day test with my simulated data. I would pad that by a few to be a bit conservative.
