Typically CR is set up so that for a party of four level 4's then a CR 4 encounter would be a moderate encounter. Understand that CR tends to break apart at higher levels usually levels 12+ due to 5E characters essentially being super heroes from the get go and it gets worse at higher levels.
I'd stick with @Farling's answer as a starting point. Running through the DMG is analagous to basic math for Encounter Balancing. There are other methods out there to determine the probability of the encounter's outcome.
Now to answer your question directly: As far as your DM-heart desires.
You *can* pit your 3rd level party against a CR 15 dragon. But what encounter building in the DMG is trying to tell you is that there is a probability of difficulty and a probability of outcome if you stay within a set of parameters. If you do throw a CR 15 anything at a 3rd level party, the safe money is on a TPK. Unless, of course you pull the monster off of them as a way of showing them the BBEG and a goal to work towards.
Some might say that an exciting encounter/adventure makes the party feel like they have a 70% chance of PC death and only a 30% chance of success when in reality, they have a 70% chance of success and only a 30% chance of PC death.
Once a DM gets to know their party, this becomes very easy to fine tune, even at higher levels.
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“Travel is fatal to prejudice, bigotry, and narrow-mindedness, and many of our people need it sorely on these accounts. Broad, wholesome, charitable views of men and things cannot be acquired by vegetating in one little corner of the earth all one's lifetime.” - Mark Twain - Innocents Abroad
In practice using the DMG method of encounter balancing won't particularly challenge the party unless you actually go through 6-8 encounters per day, which is hard to arrange outside of a pure dungeon crawl. Most DMs go with a smaller number of Deadly encounters.
The above is all helpful, and should provide a good starting point.
That said, CR is more of a guideline than a hard-and-fast rule, as each party is different. Factors like player skill, optimization, and class choices can drastically change how dangerous an encounter might be to a party and that cannot really be measured by CR alone. Part of any campaign is going to be trial-and-error. You should toss out a few encounters that are too easy and too deadly for your party (and either fudge some of the numbers to ensure they survive or add some other mechanism, like reinforcements or the enemies being driven off by something) to ensure your experiment does not result in a TPK. Playing around with the CR balancing early can give you a good idea of your baseline for that particular party makeup, allowing you to extrapolate other CR needs based on your particular party, scaling up the difficulty as they level.
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I know it says that a CR3 creature is a challenge for 4 lvl 3 players, but how many CR can you go over player level
See https://www.dndbeyond.com/sources/dmg/creating-adventures#CreatingEncounters for an explanation of CR vs party level
Encounter builder is the easiest: https://www.dndbeyond.com/encounter-builder
Typically CR is set up so that for a party of four level 4's then a CR 4 encounter would be a moderate encounter. Understand that CR tends to break apart at higher levels usually levels 12+ due to 5E characters essentially being super heroes from the get go and it gets worse at higher levels.
I'd stick with @Farling's answer as a starting point. Running through the DMG is analagous to basic math for Encounter Balancing. There are other methods out there to determine the probability of the encounter's outcome.
Now to answer your question directly: As far as your DM-heart desires.
You *can* pit your 3rd level party against a CR 15 dragon. But what encounter building in the DMG is trying to tell you is that there is a probability of difficulty and a probability of outcome if you stay within a set of parameters. If you do throw a CR 15 anything at a 3rd level party, the safe money is on a TPK. Unless, of course you pull the monster off of them as a way of showing them the BBEG and a goal to work towards.
Some might say that an exciting encounter/adventure makes the party feel like they have a 70% chance of PC death and only a 30% chance of success when in reality, they have a 70% chance of success and only a 30% chance of PC death.
Once a DM gets to know their party, this becomes very easy to fine tune, even at higher levels.
“Travel is fatal to prejudice, bigotry, and narrow-mindedness, and many of our people need it sorely on these accounts. Broad, wholesome, charitable views of men and things cannot be acquired by vegetating in one little corner of the earth all one's lifetime.” - Mark Twain - Innocents Abroad
In practice using the DMG method of encounter balancing won't particularly challenge the party unless you actually go through 6-8 encounters per day, which is hard to arrange outside of a pure dungeon crawl. Most DMs go with a smaller number of Deadly encounters.
The above is all helpful, and should provide a good starting point.
That said, CR is more of a guideline than a hard-and-fast rule, as each party is different. Factors like player skill, optimization, and class choices can drastically change how dangerous an encounter might be to a party and that cannot really be measured by CR alone. Part of any campaign is going to be trial-and-error. You should toss out a few encounters that are too easy and too deadly for your party (and either fudge some of the numbers to ensure they survive or add some other mechanism, like reinforcements or the enemies being driven off by something) to ensure your experiment does not result in a TPK. Playing around with the CR balancing early can give you a good idea of your baseline for that particular party makeup, allowing you to extrapolate other CR needs based on your particular party, scaling up the difficulty as they level.