Everyone who has done composting would probably agree small particle size of compost ingredients is essential for quicker composting. Is there any way to shred compost ingredients without using machine (shredder/lawn mower)?
Well, it's a machine but not one you specified - leaf blower. The ~$80 Corded models for Toro and Worx have vac attachments and metal impellers. I use them in the fall for my maple leaves. They'd work fine for kitchen scraps. (To anyone who might read this - use a 12 gauge cord, and the gutter cleaning attachment is a gift from heaven).
Dry leaves you can just walk around on.
Honestly though, if my carbon pile wasn't on the other side of my lot from my maples, I don't think I'd shred them anymore (the vac attachment to my blower shreds, and vacuuming I to a bag is a lot faster than blowing 500+ cubic feet of leaves 100ft to the pile). The gain in decomposition speed from shredding is minimal if you turn properly and have a bit of an aeration element in the compost. (I break apart a stick or two, or use loppers to cut it if it's green - only need a handful. Doesn't matter if it's not rotted at finish BC it's there for aeration).
So the cheapest way to shred waste is to not do it. You'll be tumbling/pitchforking it 5-8 times as it rots, and that's enough. When you sift the final product, anything too large goes right back in the pile anyway. I think you'll be surprised by how little difference there is. Handful of inch long 1/8 - 1/2 inch diameter aeration sticks and a handful of dirt for every ~3 gallons of compost is how I do it.
I'm a fan of http://www.vegetable-gardening-with-lorraine.com/homemade-compost-bin.html - but a black locking-lid plastic garbage can with holes drilled all over is just as good, less work to make, and a lot less work to tumble. And a barrel is better for short term applications like potting mix recyc.
Wow I went off-topic. Actual answer summary: cheapest way to shred waste is don't do it, just turn/tumble your compost properly.
There is a way without using any powered machine, but it's very time consuming and not terribly efficient; a pair of loppers and sharp secateurs and shears will be necessary. Cut the larger woody stuff into small pieces with the loppers; use the secateurs to further reduce the size, and to reduce the size of smaller twigs and stems; use shears to chop through soft stuff. It won't be as effective, and it will take some hours if there's a lot, so it's only worth doing if someone has absolutely nothing else to do, or finds it therapeutic. Personally, I'd prefer to watch paint dry, so I'd rather just chop up the big stuff and wait longer for the resulting compost...
Other than ripping stuff that can be ripped by hand (most of which does not particularly need to be, as it will break down easily anyway) any shredding is going to involve a machine, unless you can find an animal that wants to eat your compost ingredient and thus "process" it for you.
Some people do, in fact, use a flock of chickens this way, but it's a roundabout route to composting. They will eat and to some extent shred with their feet (scratching) and excrete processed material (manure) onto the stuff they don't choose to eat. But they won't have much interest in (say) wood waste unless it's crawling with bugs (and it's the bugs they will want, in that case.)
Otherwise it's going to be some kind of machine - Bamboo's loppers and secateurs are machines, Paul's leaf blower is a machine, various mechanized compost processing devices that range from the well-established and expensive but reliable (hammer-mills) to well-meaning attempts to make "low-tech" machines that may not work so well (I've found a trip-hammer machete with a chute to feed it and a cast concrete grindstone in my surveys of this subject in the past. Both looked like a lot of work to use. I can't currently find either one on the web.)
In the very low-tech line, this is described as a compost chopper, but it's still a machine (albeit a simple one) - that looks like a lot of work to use.
Alternatively there is always time, as Paul & Bamboo both mention. I like to help time - I have, via experience, decided that I really don't enjoy feeding a brush chipper (or listening to it, or trying to make sure that I end the day with as many arms/hands/fingers as I started the day with.) I have found that I can markedly increase the rot-rate of a pile of brush by encapsulating it within a pile of horse manure, which maintains a moist environment and supplies some nitrogen. The hugelkulture folks simply bury the woody stuff under the planting beds and let it rot in place, which also works. Another way to use woody material that breaks it down is to turn it into charcoal.
Anything that was once alive is IMMEDIATELY attacked by decompose-rs. They do all the work for you, all you need is patience, add nitrogen to the compost to feed the decompose-rs, keep the compost turned once a week so the decomposition is aerobic not anaerobic (p u) and protect from critters. While you are cleaning up debris the more time you have to break sticks into little pieces the better...if you are in some kind of hurry. I make different piles; one for herbaceous stuff and food scraps (never meat) and one for woody debris. I add more nitrogen and leaves and other herbaceous stuff to the woody debris and keep turning both. Takes a bit longer, but really not that much...months not a month as it takes for the herbaceous stuff. Moisture, nitrogen (there used to be a kitty litter made from alfalfa that was cheap and worked extremely well) and air...oh, I also added soil into both because that helps inoculate your piles with even more decompose- rs.
I've had some success using a hedge trimmer on a pile of cut (thin) branches, brmables etc., though the aim was mainly to compact them rather than to speed up composting. You can even do this in the bin to save having to tidy up so much afterwards.
For brittle material (like dried-out canes) you can chop downwards with a spade, so long as your spade has a fairly flat end.
In fact, depending on how your composting is structured, you may find that shredding too finely impedes drainage. I use a dustbin (actually 2 of them) with drainage holes added, and the lowest holes block quite easily unless there's a twiggy layer at the bottom.
In the compost bin I put the old dead tomatoes vines and lay them across all in the same direction and then take the chain saw to them cutting them up into 4 inch lengths, this hells them compost especially by putting on top of the cutting lots of tasty vegetable compost cuttings from the kitchen.