to leave running all day?
Rather than try to come up with the two devices that are the MOST and LEAST expensive right off the get-go, how about if we compare devices, then eventually the two that fit the OP should surface.
TV?
AC?
FM Radio?
Night Light?
Laptop?
Vacuum?
100 Watt Bulb?
Oven?
Others?
I think the big factor is how much heat is generated. So presumedly, an electric heater uses a lot of electricity.
I think a large air conditioner window unit on a hot summers day would could out near the top.
One of those plug-in air fresheners would rank at the bottom (for electricity use).
It is current that you’re paying for. (Amps)
Typically, the highest current drawing appliances in your home would be
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An electric furnace. (usually associated with a heat pump) That includes electric radiant panels etc. In a good size house it’s nor unusual to have this appliance on a 60-100 breaker. That’s a lot of juice.
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An electric dryer. (typically on a 30-40 breaker)
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An electric stove. (30-40)
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A central air conditioner running outside. (30-40)
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Misc power tools may draw a lot of power—table saws, air compressors etc
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Pool heaters
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Excessive outdoor lighting
Theres more for sure, but that’s the common ones.
The lowest power consuming device I have is an electroluminescent night light. It’s rated at 115 VAC, 0.03 W.
One other thing…
None of the high current items would “run all day.” They are all controlled in one way or another.
But the HVAC equipment will probably see more “run time” than your dryer/stove etc. The HVAC equipment is the most expensive electric cost in most homes.
Of the stuff you would leave on all day, I would imagine high wattage bulbs, like 100-150 indoor bulbs/floods would use a fair amount of juice if left on all day. I’ve converted to compact fluorescents where I could.
If you left the burners on, an electric stove’s not controlled at all. The oven is controlled by thermostat, so it would cycle on and off. The receptacle for an electric stove in my house is on a 50-amp breaker, so it’d be capable of drawing some serious steady power.
Everything else is on a timer (dryer) or thermostat control (furnace, air conditioner) so my vote for the thing that has the most potential to suck power would be the stove.
The 50-amp breaker for a stove is also the largest breaker in our panel. The central air has a 40-amp breaker, and the dryer is on a 30-amp. (Happily, I cook and dry with gas - the house was built for dual-power in the laundry and kitchen.)
I’m thinking the most expensive electrical device would be the Hoover Dam.
Running a close second is the Pentium IV in my home office. It’d keep the room warm all winter long. In my desire to help find ET with SETI-@-Home, I swear it added $25 to the power bill.
That was pretty much my answer. The biggest electricity “pipe” goes to the stove. Roast a turkey for a few hours and you’ll see a noticeable spike in your electric bill.
As for least expensive – I’ve got a clock that will run over a year on a AA battery. And watches that consume even less power (but the batteries are more expensive). Certainly no one ever proposes conserving energy by turning the clocks off when you leave a room 
My friends just bought a house; the previous owner threw in all her pottery-making equipment.
My friends therefore now own an electric kiln, which looks a lot like this. I believe it requires a 50- or 100-amp line. The plug has four prongs connected to the thickest power cord I’ve ever seen outside a resistance-welding shop.
I think they’re afraid to turn it on.
I have a friend who used to own a fair-sized kiln. He said a firing used about $40 of electricity.
It’s actually watt-hours: current * voltage * time.
That’s pretty phenomenal. How much gas (in dollars) would a gas fired kiln use in a firing?
It’s controlled by a human being!
Just as the oven is controlled by a thermostat—and ultimately turned off by a human being when the food is done---- the burners are turned (on and) off by the human when the food is cooked to satisfaction.
My answer had a certain practical assumption to it----that the burners on your stove would not be turned on intentionally and left on all day long. Nor would the oven be left on all day long. (even if being cycled on and off by the thermostat)
OTOH, an electric furnace is in fact “left on” in as much as it is allowed to turn on and off all day long based on the thermostat setting.
So an electric furnace—which often will have 15-20,000 watts (or more) will see much more run time during the winter (in a cold climate) than your stove. )(unless you’re a caterer)
So it’s not the pipe. It’s how often there is current running through the pipe. If your stove “runs” 1 hour a day, and your electric furnace runs for the equivalent of 5-8 hours a day on a cold day, I guarantee that the electric furnace will use more KW hours than the stove.
As Xema pointed out, it’s KW Hours that you pay for. It’s not uncommon to have a 50 amp breaker for a stove. (although the breaker may be oversized) It’s equally common to have an electric furnace with a 20KW heater on a 100 amp breaker.
Most electric furnaces will draw more current than an electric stove. But once you add the element of [run] time, the HVAC equipment will cost you more. (assuming a cold climate, of course)
I always thought a vacuum would be fairly high on the MOST expensive list, that’s why I rarely use mine. 
But seriously, I think I have blown more fuses with a vacuum than any other device. But then again, who would leave their vacuum running all day?
Does it cost most to play your music loud? I would imagine so.
Yeah, but re-read the OP. The question was, “what’s the most expensive to leave running all day.”
Yeah, but re-read my response.
Left running all day, a 20+KW electric furnace will draw more current than a stove.
(It’s not entirely clear to me that the OP wasn’t asking for a practical answer; (vs a literal reading) but either way, an elctric furnace will cost you more than the stove)
What got me thinking about this is my radio that sits next to my bed. I tend to leave it on all day, forgetting to turn it off in the morning.
Leaving your radio on every day, 24 hours a day, all month, wouldn’t hardly be noticed on your electric bill. Pennies a day…