The crafting system in Elder Scrolls Online uses an upgrade system where you can apply “tannins” (or stitchings, depending on the crafting skill) to your items for a chance to improve them. The more tannins you try to use in a given attempt, the higher your success probability.
Because it was a lot more fun than actually playing Elder Scrolls Online, I worked out the optimal tannin-applying strategy for different models of marginal upgrade utility. The details are written up here.
I expect that the normal situation is to have diminishing marginal utility–your second success is less helpful than your first success–because you can always choose the order of items you upgrade to start from the most helpful to improve. In this case, you should use as many tannins at a time as you can to make it more likely that you get any successes.
If each success is equally valuable, then your strategy doesn’t matter at all: the expected utility is (probability of success with a single tannin) * (# of tannins) * (utility of a single success), no matter how many tannins you use at once.
If for some reason you have increasing marginal utility (perhaps there’s some perk whose benefit is super-linear in your armor rating?)–that is, your second improved item provides you even greater benefit than the first improved item–then you should be “greedy” and use one tannin at a time to maximize the chance of getting a larger number of successes.
When I did some initial Googling to see if this was a solved problem, I came across a very useful tidbit. The next time you need to make $10 million by midnight on the roulette wheel, your best strategy is the “bold” strategy in which you bet your entire fortune each round. [Some more discussion here.] Of course, if you just want to maximize your expected wealth, then your best strategy is the “don’t play roulette, dumbass” strategy.