I Have A Huge Bottle That I Put My Coins In. I Was Wondering If I Was To Choose To Fill It Completely Up, What Would Be Worth The Most When It Is Full?

You have a very interesting question.  I've wondered about this myself.  It's an even more interesting question with a lot of probability and statistics issues if the coins were an assortment of pennies, nickels, dimes and quarters.  It involves how well coins -- shaped like flat little cylinders -- pack into a volume.  And if it was a question of weight, rather than volume, then it includes a rather odd random variable, in what decade was it filled?  Years back the materials used in coins changed to a copper-nickel sandwich (making them cheaper to manufacture), and the stamping impressions have changed, so old coins don't weigh the same as newer coins.  But let's first assume these are all modern coins and not the ones that are mostly in the hands of coin collectors.

Your first question asks what your bottle would be worth.  If you take a similar bottle and fill it with water using a measuring cup, you can determine how much volume it has.  We'll need that number to proceed if we want to know the total value of coins in your bottle.  Then we would need to estimate a packing density of coins, which will be a number less than 100% (probably closer to 60% since they are round and even though they tend to lay in flat layers, they can never fill in all the gaps).  But for now, let's address your "for instance" case of an arbitrary volume full of quarters versus dimes.

We'll trust the mint for coin specifications and use their values;
www.usmint.gov

A dime has a diameter of 0.7 inches (1.8 cm) diameter, and is about 0.06 inches (0.13 cm) thick.  Dimes have a volume of 0.33 cm3 and weigh 2.27 grams, giving a single dime's volume a density of 2.27 grams per 0.33 cm3 or 6.85 grams per cm3

Quarter are about 0.95 inches (2.4 cm) diameter and are 0.055 inches (0.14 cm) thick.  They weigh about 5.67 grams and have a volume of 0.63 cm3 so this is a density of 8.96 grams per cm3

But a quarter's face value is 25 cents versus a dime's value at 10 cents, so they are 2.5 times more valuable per coin.

The volumes are 0.63 cm3 per quarter and 0.33 cm3 per dime which is a volumetric ratio of 1.91 dimes per quarter, or if we invert it, 0.52 quarters per dime.

So from this, we can get 2.5 times more face value with quarters but a coin volume difference of less than 2.5, making quarters more valuable per unit volume.  In other words, since it would take two and a half dimes to make the same monetary value as a quarter, and two and a half dimes are 0.52 the size per dime times 2.5, they are equal to 1.3 times the volume.  Therefore, dimes are a 30% increase in volume.

A possible flaw I see is if this analysis is if the bottle is small enough to prevent quarters and dimes to lay in the same packing ratio.  If the width of the bottle is small relative to the diameter of the coin, then it's a different problem entirely.  With coins the size of dimes or quarters, 5 cubic inches would not be 'huge' enough, but 5 cubic feet certainly would be.
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I believe your question is: If you were to fill the same huge bottle completely with either all dimes or all quarters, which one would be worth more?

My guess will have to be the one with the dimes, since they are thinner and much smaller and therefore more of them will fit inside the same space. This is like comparing a bottle filled with sand or beans: There is a lot more wasted space with beans. However, a quarter is worth 2 1/2 times as much as a dime. So the result may actually be quite close, one way or the other. It would be interesting to find out by actual experiments.
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When your huge jar of coins is a quarter full...count it..now times it to 4..now you have a idea of what the amount would be in the jar (or) when the jar is half full..count it..now double that amount...now you have a set amount of what it would be...because you know what you throw in the jar every day or pay day of the week so it probably is the same amount of money (that you throw in the jar) every time you add your change to it...use a maker if you need to to mark your jar...half full...quarter full...and if you want to know which coin has more value...fill the coin jar till the 1/4 quater line mark is reached of each  the dimes and quaters seprately...now times that to four...djoyful
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We don't know how wide or tall the bottle is. I understand that "huge" is an adjective, but huge could mean anything. The best solution to your problem is putting aside some time to count all of your money once the bottle is full. =)
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Anonymous commented
What difference does the size of the container make? If it is small, lets say 5 cubic inches if it is worth more filled with dimes than quarters the same will be true if the container is 5 cubic feet. It will just be worth a lot more.
Anonymous commented
In the case of 5 cubic inches, that would not be considered "huge" relative to the size of the coins. 5 cubic feet certainly is.
Dimes most definitely I just cashed out a hundred and fifty dollars worth just in a one gallon detergent jug...I now separate them into their own jugs for easier rolling to take to the bank....
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Robyn Rothman commented
Don't be so sure about that. It took 1,500 dimes to make up \$150, but it would have only taken 600 quarters. 