8th order butterworth frequency response

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.

@Confused: Sorry - but I have to start with some general comments:

In principle, for an 8th order filter you have two basic alternatives:

(a) Direct realization (active topology derived from a passive and tabulated reference structure, and (b) Cascade realization as a series connection of 4 second-order stages.

I suppose, you are following the latter approach - and here you have again several alternatives (how the various 2nd order stages are realized). It seems that you have decided to use Sallen-Key realizations because you have mentioned finite gain values.

But also in this case, you again have alternatives: Unity gain approach, gain-of-two approach or equal-component approach (with gain values lower than "3"). Independent on these 3 alternatives, you must know that all 4 stages look different: They are individually designed for equal pole frequencies (applies only for Butterworth response) but for different pole-Q values to be found in filter tables. Hence, you will NOT have 4 identical 2nd-order stages but each of the 4 must bedesigned separately.

I am not sure if this answers all of your questions - perhaps it helps if you could give us some more detail of your envisaged design.

EDIT 1: The following link leads you to a document (from TI) which gives you the Q values for your 8th-order filter on page 8 (correction: page 9)

http://www.ti.com/lit/an/sloa049b/sloa049b.pdf

EDIT 2: For your convenience, here are the formulas for designing the 4 stages (equal pole frequencies wp, different Qp values) - to be applied for the gain-of-two version:

C1=C2=C and wp=1/[C*Sqrt(R1R3)] and Qp=Sqrt(R1/R3) with R1: Most left resistor(connected to input signal). For a gain of "2" you can use any two equal resistors in the negative feedback path.