I have been following here, but haven't commented because I felt like I didn't have much to offer. I STILL don't have much to offer, but have time to chat, so here goes!

And sorry if you are saying DUH to a lot of this..... just thinking out loud here.
For anyone looking into this, try googling (or searching on Pinterest) things like "emergency food storage" and the like. If you are storing a lot of canned goods there are some pretty interesting plans out there for big racks to use to store the cans.
BE SURE you are rotating your stock ALL THE TIME! Oldest to the front, newest to the back..... or you will end up with 10 cans of beans that expired 5 years ago!
Be sure your storage area is DRY and not terribly hot or cold. Stuff will go bad MUCH FASTER if conditions aren't good. And if you are buying dry goods like flour/grains/oatmeal/cornmeal, you may want to be sure you have FREEZER space so that it doesn't get buggy or rancid.
Be sure to know your "audience"..... in looking at the pictures/lists of what others here are buying, I realize that there is probably no "perfect list" that would be good for everyone. I never buy/eat what some of you have mentioned buying/stockpiling and I imagine I eat things you wouldn't..... so you need to tailor your list to suit YOUR FAMILY.
Take into consideration WHY you are buying this. Is this food you would plan to have on hand in the case of a major catastrophe? If so, does it need water/energy to be eaten? And if so, where would this water/energy be coming from? If it is food for just an economic emergency (unemployment for example) is it food that your family ACTUALLY EATS? Sure, we all KNOW we could survivor on brown rice, black beans, and oatmeal if things were desperate..... but no sense in stocking up on this stuff if your family refuses to touch it!
One of you mentioned a thought that reminded me of something I learned from the Tightwad Gazette years ago (BTW, if you are interested, look into finding her books.... a bit dated at this point since they are from the early 90's, but interesting reading still..... written by Amy Dycyczyn). If you know your family uses one jar of peanut butter a month..... and that peanut butter goes on rock-bottom sale once every 4 months, then you only need to buy a 4 month supply at a time. But KNOWING that sale cycle takes some time and research and record keeping.
Because I live in a pretty rural area (30+ minute drive to the nearest grocery store) I have always had a back-up grocery supply on hand..... but most especially when the kids were little and we didn't NEED to leave town very often (as they got older and involved in sports outside of town, I had an excuse to hit the market more often). It took a while to figure out how to shop just once every 3-4 weeks! I learned what can be frozen (milk, butter, even eggs in a pinch, any fruit/veggie intended for a smoothie, beans, cooked rice, cheese, deli meats, cookies, hummus), what produce lasts in the fridge (citrus, apples, carrots, onions) and what needs to be eaten the same week it is bought (greens, broccoli, grapes). We would start our months eating salads and lots of fresh stuff.... and then by the end of the month we were eating casseroles made with frozen and canned stuff and more pasta and rice.... NBD!
Since those years DH and I have both lost a significant amount of weight and as such we eat MUCH HEALTHIER than we did when the kids were smaller (although we have always eaten fairly healthy) and we "require" fresh produce more often.... but DH goes to a gym near the market a few times a week, so he can stop as needed.
Anyhoooo...... I LOVE the OP's idea of using just $10/month to get a stockpile going! I've been "stockpiling" to some degree for years now, and there are definitely things I've learned that we need MORE of (canned tomato products, especially in the winter), grains/beans (quinoa, barley, popcorn, lentils, couscous, brown rice), organic natural peanut butter, salsa, .... and things we can do without (anything processed, pasta [DH eats low carb so by default we all eat pretty low carb], canned veggies [don't like most of them.... would rather have frozen], canned soups [don't like most of them and don't care for the ingredients], baking supplies [I don't bake], cold cereal [we don't eat it])...... But again, I LOVE the idea of using $10/week to start rebuilding my recently depleted stockpile! can't wait to share my next "shop"!!......................P