Forgot a toothbrush, paste, razor, shampoo... ask the front desk of your hotel, they usually have extra basic toilitries for free.
Chapped lips? Dry legs? Go to the hotel lobby or local Starbucks and grab a packet of honey along with a packet of sugar (many packets of honey and sugar if it's for your legs) and you have a DIY exfloliator when mixed. Scrub where needed and rinse off. You should use enough sugar to honey that it makes a thick paste like in the photo below.
The conditioner provided next to your lotion in a hotel bathroom can be a shaving cream substitute. You can also mix conditioner with honey and any natural oils you can find around the hotel kitchen nook (olive oil usually) and leave on dry hair for 15-30 minutes for a quick deep conditioner.
If you have nail polish on hand, you can paint wet nail polish on top of old, dried, chipped nail polish to remove it. Just paint on top of the old nail polish and wipe with a cotton ball or wad of toilet paper if you don't have a cotton ball.
An oldie but a goody, use toothpaste in a pinch to dry up a zit.
Revive a dry, tired face by brewing tea (caffenated green tea is best) and wiping your face with the warm tea. Place the wet tea bags on your eyes to reduce dark circles and puffiness as well.
A little bit of hand soap or shampoo can be used as a facewash if you're really in need, but be sure to moisturize afterwards. Always try just water and gently scrubbing with your fingers first.
I hate when I forget nail clippers and really need to cut a jagged nail. To hold you over from going crazy, a pair of scissors can clean up the edge. Again check with the hotel front desk if they have a file as well.
What other things do you commonly forget on trips? Let me know in the comments and I'll see if I can't find something to DIY.