Does a pan used for cooking scrambled eggs multiple times a day need to be washed with soap every time?
Bacteria have several requirements to grow — water, food/nutrients, oxygen/lack of oxygen (some bacteria), temperature, and pH. The acceptable range of conditions varies, but if you deprive them of one of their requirements, they don't grow. We use all of those in food — but for your question, three are involved:


Temperature. When the pan is hot, nothing can grow. Once it cools off below ≈130°F/55°C, that changes. Until you get down to 40°F/4°C, you're in the "danger zone" where bacterial growth is possible and relatively quick.
Food/nutrient availability. When you wash clean the pan, you're removing the nutrients, so the bacteria can't grow. You propose not to do this, so this control method isn't available.
Water. Without available water, nothing can grow. This is why you want your pans to dry out after washing (even things like aluminum or ceramic, which won't rust). If there are nutrients left (perfect washing is hard), without water, nothing can grow.


So... I think as long as you make sure the pan is dry then you're fine. Wipe it out with a paper towel, you don't want any egg-gunk left (that'd be hard to get all the water out of). If there is any doubt, turn back on the burner after wiping it — heat will drive off any moisture. Just make sure not to let it get hot enough to damage the pan. (Be especially careful if its Teflon or similar).

If paranoid (which, depending on your health condition, may be justified), once the pan cools to near room temperature, put it in the fridge. You'll then have two factors redundantly protecting you from bacterial growth.