I have a big-box-bicycle that I ride around the neighborhood regularly. After trying to master the highest gear setting I now find it easy on days where there isn't a breeze, and still manageable on windy days. I'd like to ditch the lowest gear on my front cassette because I never use it -- would it be possible to move my existing gearing down and add as taller top gear? Are high-tooth-count-cogs commercially available? Do they all use similar mounting mechanisms? Looking at my my cassette I see that it looks lik each successive gear just bolted to the previous one, Maybe I could just bolt the new gear in place of my chain ring?
You can't just move all the chainrings down a notch and add a bigger one: they'll interfere with the chainstay, which makes this impossible, and it would also be difficult to get the shifting right. Also, I'm pretty sure the chainrings that will fit this crankset are only available in a small number of sizes—possibly only one set of sizes.
I'm guessing this bike has a 6- or 7-speed freewheel in back. You might be able to get one that gives you a slightly higher high gear, but not by a lot—the small gear is probably 13 or 14 teeth; you might be able to find a replacement with a small gear of 11 or 12 teeth, so (roughly) a 10% increase in gearing.
Also, as you know, this is a cheap bike. You'll find that new parts for it are disproportionately expensive.
What I would suggest instead is that you work on increasing your cadence, that is, your pedal speed. If my guess is right about the gearing on this bike, then at a cadence of 60 rpm, you're going about 15 mph in top gear. If you increase your cadence to 80, you'll be going 20 mph. If you find you can hold 20 mph easily on this bike, invest in a better bike.
Having a rarely-used grannie chainring is good if you ever strike a climb or big headwind and simply run out of legs.
Your chainrings are rivetted together, so they're not easily changed and you'd have to buy new crank arms etc.
It may be possible to change the cassette for something smaller, but if you have a freewheel then this needs an entire new rear hub, which is pricey.
Instead, look for a more modern bike, used, to add to your fleet. You want something with a cassette on the rear wheel, and 8/9/10 separate cogs on the back gives a lot of options. There would likely be two front chainrings (a double) rather than a triple like you have now.
Generally, big-box store bicycles aren't worth upgrading. If you get to the point you're riding enough to want an upgrade, it's usually a lot more cost-effective to just buy a better bicycle that meets your needs.
Importantly, some of the best values can be found in used bicycles. See What should I look for when buying a used bicycle?
Also, once you get to the point of riding a lot, you need to buy a bike you like to ride. Be really careful spending a lot of money on something like a "hybrid" bike - such bikes won't be as good off-road as a good gravel or mountain bike, nor will they be as good on roads as a good road bike. If you spend money on a bike you don't like to ride, you'll likely find that you've wasted that money.
And yes, as stated in the answer from @AdamRice, you probably do need to work on pedaling faster. You probably have a 42-tooth big chainring, and likely a 12- or 13-tooth smallest cog on the rear. On a 29er mountain bike, you'll reach over 21 mph on a 42-tooth front chainring and a 13-tooth cog at 80 RPM per https://www.sheldonbrown.com/gear-calc.html
Most cheap bikes use a square taper fitting for the cracks and any crank with the same square taper fitting will fit. There are thousands of such cranks available - just search eBay for square taper crankset to see a bewildering array of options. But as other answers have mentioned there are potential problems.
With the crank you have the three chainrings are riveted together and cannot be independently changed. However you can buy cranksets that allow the rings to be changed independently. I have an SRAM Truvativ S650 crankset that allows the chainrings to be changed independently. Or given the huge number of cranksets available you can probably find one with exactly the ratios you want.
The other answers have described the potential problems you can run into, but entry level cranksets are not expensive and it can be fun to experiment so if you don't mind spending (and possibly wasting) a bit of money I'd say give it a try. I did exactly this on my bicycle (see Can I (easily) change this chainring?) though that was with a single speed crank, and I'm very happy with the larger chainring I managed to fit.