Cut CeleryCredit: @LiviBui
If you're like most people, you toss the ends of your vegetables in the trash.  Well, did you know you can regrow vegetables like green onions and lettuce from those scraps?  
Easy - Quick Growing in Just Water
Green Onions and Scallions
Green onions or scallions, give a great bite when garnished on soups or salads. Normally, when cooking with green onions or scallions, only the green parts are used while the white parts are discarded.  Simply placed the trimmed white portions in a cup of water, put the cup in a sun-lit area and watch them shoot up like weeds.  Be sure to change the water every few days or when the water begins to look murky.  These plants can grow as fast half an inch to an inch a day so, you'll never have to buy green onions again!
Green OnionsCredit: @LiviBui

Have a hankering for a salad?  Why not grow your own? When using whole heads of lettuce, cut off the base and place it in a shallow dish of water in area with sunlight.  In a few days the base will start to sprout.  Lettuce does not grow as fast as green onions but are similar in that you can keep trimming what you need and it will keep growing. This is a economical and convenient way to have a fresh side salad whenever you want.

Medium - Slower Growing and Need Soil
Similar to lettuce, take the base of the cabbage and put it in water.  Once the roots and leaves start to grow transfer the root to soil.  Leave plenty of room between cabbages as they can grow to be large but do not let it grow too large as it starts to lose its flavor when it grows too large.
Bok Choy
Bok Choy is a Chinese cabbage used in various stir-fry dishes.  Similar to lettuce, cut off the normally discarded base and place in water.  After a few days new leaves will sprout.  
Similar to green onions, chop off the roots of the onion leaving about half an inch of onion and soak the roots in water.  After the roots start to grow, transfer the roots and cover with one to two inches of soil.  Water as needed and keep in a sunny place.  Once the onion has regrown, you can harvest and repeat.
Before removing the stalks, cut the bottom off and place in water.  As previously mentioned, change the water as it starts to become discolored. You will see celery growth within a few days but to get full stalks will take months.  CeleryCredit: @LiviBui
Lemongrass has a citrus flavor that brings brightness to Thai and Vietnamese cuisine. To regrow lemongrass cut off the tops of the stalks and place the remainder in a tall glass of water.  Make sure the tops of the stalks are not covered by water.  Change the water every few days until roots form.  After the roots have grown for a few days transfer the stalks to soil and place in a sunny area.
Hard - Want a Challenge?  Try these...
Cut the nubs of ginger with the skin and soak it in water overnight.  Plant the chunks of ginger in a pot of soil with the nub facing downward.  Water and place in an area with indirect sunlight.  Ginger makes a beautiful houseplant or when ready, just pull the sprout to harvest the ginger.  
Sweet Potatoes
When the sweet potato starts to grow eyes, cut the potato into two-inch pieces around the eyes.  Leave the sweet potato out overnight and then plant the sweet potato chunks in a pot.  Spacing is not important since they will be replanted.  When the sprouts reach four to five inches, replant about nine inches apart. It should take a few months before the sweet potatoes are ready for harvest.
Yes, this is a fruit but I thought this was so neat, it had to be included.  Regrowing a full fruit is highly unlikely, but you can grow a beautiful pineapple house plant.  Simply cut the top of the pineapple, remove all the remaining fruit, plant it in soil with the top sticking out, water and place in a sunny area. 
Next time you think about throwing away vegetable (or fruit) scraps, consider planting and regrowing them instead.  Do you have vegetables I've missed or ones you've tried to regrow?  Leave a comment and tell me about it!