If you only had 100$ a month to spend on food...

Lots of eggs and frozen vegetables and any meat on sale.

No drinks, just water.

And if you live by where people have farm stands or co-ops, you can get eggs pretty cheap from them and sometimes the co-ops give you great deals for stuff throughout the summer.

I used to feed my whole family on $100/month and do it pretty well, but prices have gone up...

I'd buy high-quality eggs, I'd probably look for ways to buy good meat in bulk (buying a cow share or something like that).
Barring that, I'd get whole chickens so I could parcel them out and make soups from the bones. Stew meat to stretch with lentils/grains

Lots of lentils and beans. Whole grains like barley and oatmeal.

Greens - frozen if necessary.

Store-brand flour, make our own bread (already do this: most affordable way to get quality bread without additives).

I'd stretch meat in stews, with veggies and legumes.

Lottery tickets

Use some of the money to buy 3-5 Sunday newspapers on a weekend where there are a lot of inserts. The coupons you get will help you extend your budget greatly.

Bulk chicken
A whole pork tenderloin (cut into portions, freeze and cook when needed. Can be turned into ANYTHING.)
Bulk beef, and celery, carrots, + old fashioned oats to mix into the beef to "bulk it up"
Oatmeal mix-ins (for the old fashioned oats) like brown sugar and raisins
Eggs
Bulk, DRY beans (you save serious moolah buying them dry)
Bulk brown rice
Bulk almonds, or any other nut
Peanut butter
Protein powder - you can mix this into ANYTHING and feel satisfied
Milk/Almond milk/Soy milk/Etc
Dry milk (for those days you run out. Not too yummy to drink straight, but comes in handy if you have a recipe that needs it and you're fresh out)
Bulk whole wheat pasta
Frozen veggies (Anywhere from .89-$1.00 a bag. None with cream sauces.) Green beans, peas, broccoli, asparagus
Fresh fruit seems to be cheaper than frozen most of the time. "Cheaper" fruit for me tends to be bananas, apples, pears, peaches, etc. I try to avoid the canned whenever possible, but the ones in fruit juice can be cheap if you're in a rut
You already have celery and carrot sticks in this list. Red bell pepper is good, too, but a little pricy for what you get.
Bulk heads of romaine lettuce. Pre made bags of salad are almost worthless for me...I eat too much salad and that seems to go bad FAST.
You can make homemade hummus, if you buy the $7.99 jar of tahini. It lasts for like...6-8 batches of fresh hummus. If you don't feel like it, just buy the big thing o' hummus.
Bulk greek yogurt. The little cups add up over time.
Whole wheat bread
Cheese
Sliced low sodium deli meat
A block of good cheese. Can be cut into 1 oz portions. Cheaper than string cheese/pre cut cubes
If you have a well-stocked pantry, MAKE your desserts. Buying those little 100 calorie things/bars and such really adds up. Make protein bars! I ALWAYS have chocolate baking powder and honey on hand.
Whole wheat crackers

You can find recipes for pretty much anything. Look up Budget Savvy Diva. She has some nifty stuff.

Some of this stuff is meant to be stretched over time. If you buy in bulk, you'll be amazed at how long things last.

If you have an Aldi's where you are, for the love of god woman SHOP THERE. So cheap, and AWESOME. Best prices on fresh produce, hands down.

Shop when things are on special. Ask your store's butchers when meat goes on sale and buy it up when you can. Use coupons. Check your stores' ad paper and create a list at home so you can tally the prices on things you know. Takes the guesswork out of everything.

I've been here, living on $100-120 every month. It's possible!

ETA: Soups, homemade 'Hamburger Helpers', Asian food nights (rice, veggies and protein), veggie burgers (homemade), steamable tv dinners for the lazy nights with a nice salad, taco night with homemade tortillas (buy your cornmeal!), pasta night, 5 Ingredient Fix crockpot meals...I live on these already. Cheap but effective.

Find a 5 Ingredient or Less blog, cookbook or whatever. It's ingenious.

More than half the people in the world live on less than $2 a day for food.

Rice and beans.

Whole coffee beans
Bulk carbs (whatever's on sale, but likely quinoa, couscous, millet or wild rice)
Any fresh vegetables in season/on sale (supplemented by veggies from garden)
2 dozen eggs
2-3 blocks of tofu (Can find 16 oz blocks on sale for 99 cents in our town. Or 2 whole chickens--cheaper to cut them up yourself, and you can make stock with the carcass)
Bulk dried beans
Bag of whole wheat flour to make bread, pizza dough, etc.
Rolled Oats

I would do bulk oats, lots of eggs, apples on sale, PB, bananas, bulk dried beans, manager's special meat (grocery stores often put meat on clearance right before the freeze/cook by date) and frozen veggies.

Also Bobs Red Mill grains are sold pretty cheap at Ocean State Job Lot and Big Lot (at least in my area) and you can get really good deals on bulk grains to make your own food.
Processed stuff is always more expensive. Learn how to make your own breads and stuff like that and it will save a lot of money.

protein drinks, eggs, yogurt, frozen chicken, frozen veggies, big bag of apples. I think I could get all of that at Costco for $100

that's about what i spend on food a month anyway - i feed just myself though. i'm a vegetarian.

i go to the farmer's market every weekend and load up on fresh fruits and veggies and eggs.
at the grocery store...

oats
dried beans
some frozen veggies
peanut butter
hummus (that's my biggest splurge item)
cheese
etc

i take the time to hunt for deals at a few different places, like publix buy one get ones.

The siuggestions are phenomenal. I got some great ideas, hope you did too. I am planning to make some items that I find pricey such as popsicles and fudgesicles. I think they will be better and certainly cheaper. Recipes are on the web. I buy only whole hot cereals for breakfast, and I am thinking of going to egg salad, tuna salad, or peanut butter and jelly for lunch. when there is a need to take out. I am not keen on ramen which is insanely cheap, but if you can find the unfried type and bulk it up with veggies it might help.

100 cups of fage total 2% plain greek yogurt!

3-18 pack of eggs, 15 frozen veggies (on sale $1 a bag), $10 worth of bananas in bulk then cut up and frozen, 10-canned albacore (I can always find them on sale for $1) Rye bread or WW Pita, Milk, Protein powder, Ground chicken (portioned and frozen) potatoes, as many fresh fruits and veggies as I could...geeze thats basically my diet right there LOL