I am baking Swedish Limpa Bread. The recipe calls for mixing rye flour with a molasses/ anise seed, salt and water mixture and letting rise overnight or nine to ten hours. Does this exhaust the sourdough starter, leading to little rise in the bread? How can I rectify this problem?
Generally when you have issues regarding oven spring at home, the problem is heat and steam in your oven rather than yeast activity. In commercial bakeries, sourdoughs are frequently fermented for overnight or longer, so that shouldn't be a problem. A lot of the expansion you see in the oven is from the moisture in your dough converting to steam before the crust has set. In a bakery, we can achieve this by using steam injected ovens which allow the crust to remain elastic longer. Bakeries also use hearth and convection ovens which transfer heat into the loaf more efficiently causing it to create steam faster.
To recreate this at home, you can try moistening the crust of your bread before baking it or trying to create steam in your oven. You can encourage steam by starting your oven at a higher temperature and turning it down after a few minutes, or by baking on a pizza stone to simulate a hearth. It's important to get the steam down first though, as the heat transfer without steam will cause a lot of crust cracking and weird rising.
It's also important to note that if your dough has a high proportion of rye flour, rye traps bubbles via gums called pentosans rather than via a gluten network like wheat dough. It generally won't rise in the same way that wheat doughs will.
I'm assuming you're not getting much oven spring from the sound of it.
Sourdough starters don't have as much oumph (its a technical term ;) ) as a commercial yeast. Even in the best case, they just don't typically spring as much in the oven (nor do they rise as rapidly otherwise). Without seeing the exact ingredients, the temperature of the environment, and some massive luck (due to the variability of sourdough starters) - its not really possible to tell you if its exhausting...but its incredibly unlikely. You're also dealing with rye which has far less elasticity than gluten.
Your options to rectify are basically either to try adding vital wheat gluten, a bit of wheat flour, or a commercial yeast. Its very common with sourdough bread these days to also add a bit of commercial yeast.