The difference between the filters you name is not that each new one invented made a closer approximation to the ideal filter, but that each one optimizes the filter for a different characteristic. Because there's a trade-off between different characteristics, each one chooses a different way to make this trade-off.
Like Andy said, the Butterworth filter has maximal flatness in the passband. And the Chebychev filter has the fastest roll-off between the passband and stop-band, at the cost of ripple in the passband.
The Elliptic filter (Cauer filter) parameterizes the balance between pass-band and stop-band ripple, with the fastest possible roll-off given the chosen ripple characteristics.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
It depends what you mean by "best possible" or "closest model". If you mean the one with the flattest response in the pass-band, you'd end up with the Butterworth filter. If you mean the best possible roll-off given a fixed ripple in the pass-band, you'd end up with the Chebychev design, etc.
If you chose some other criterion to optimize (like mean-square error between the filter characteristic and the boxcar ideal, for example), you could end up with a different design.
Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
The filters you named (Butterworth, Chebychev, Cauer) are the best, for the different definitions of "best" that define those filters.
If you had some other definition of "best" in mind, you could certainly design a filter to optimize that, with existing technology. Andy's answer names a couple of other criteria and the filters that optimize them, for example.
Let me add one other question you might ask as a follow up,
Why don't we in practice design filters to optimize the mean-square error between the filter characteristic and the boxcar ideal?
Probably because the mean-square error doesn't capture well the design-impact of
"errors" in the pass-band and stop-band response. Because the ideal response has 0 magnitude in the stop-band it's hard to define a "relative response" measurement that has equal weight in both regions.
For example, in some designs an error of -40 dB (.01 V/V) relative to the ideal 0 V/V response in the stop-band would be much worse than an error of 0.01 V/V in the passband.
Best Answer
The RC low-pass filter (LPF) is the most simple to understand intuitively.
The capacitor reacts slow to the current it gets, voltage will rise slowly. If \$V_{IN}\$ is a fixed voltage the voltage across the capacitor will rise exponentially until it reaches \$V_{IN}\$. DC is not filtered at all.
If you apply an AC signal current goes back and forth through the resistor, charging and discharging the capacitor. If this goes slowly, at low frequency, the capacitor's voltage can more or less follow the charging and discharging, and its voltage will be near the input voltage. But the higher the frequency the faster current direction changes, and the change will take place before the capacitor is fully charged. So the input voltage is no longer reached. For very high frequencies the changes in current direction are so fast that the voltage amplitude across the capacitor is only a fraction of the input.
We have a low pass filter: low frequencies are passed with little attenuation while higher frequencies are attenuated more.
LC filters are far less intuitive.
In this parallel circuit part of the current will circulate between inductor and capacitor, and the net current will decrease the closer you get to the resonance frequency. At the resonance frequency the net current is even completely zero, as if L and C aren't there.
In the same way will a series LC circuit form a zero impedance at the resonance frequency, as if there's only the resistor.