Can the Keck Observatory be improved with a third telescope? Can the Keck Observatory resolution be improved by adding a third telescope? I imagine that that would be cheaper than building a completely new telescope?
Original Overview
The Keck telescopes mainly work independently of one another. Therefore, adding a third telescope wouldn't really have much of a point (aside from the benefits of having a telescope on top of a mountain with low light pollution and excellent atmospheric conditions). In the past, this might have been a slightly more appealing idea, because the Keck Interferometer (KI) ran from 2003 to 2012, and was a combination of the two telescopes (an effective increase from two 10-meter telescopes to one 85-meter telescope).
KI had some problems, though, that prevented it from living up to its full potential. For instance, four secondary &quot;outrigger&quot; telescopes to improve the interferometer were never built, even though they were an important part of early plans. It was eventually &quot;mothballed&quot; after several of the main scientific objectives were completed. Obviously, the interferometer could be reactivated, but you'd have to make a strong case for doing so. If you could get that to happen, and get permission to build another telescopes, many might choose to use the funds for these outriggers, rather than a whole new instrument.
Finally, building telescopes on Mauna Kea is controversial because of possible environmental impacts, as well as arguments that it endangers Hawaiian cultural and religious sites. There are objections enough over the planned Thirty Meter Telescope, never mind another Keck (which would, admittedly, probably be cheaper, although the two would use different wavelengths).
In short, a third telescope would help only if the interferometer was brought back on line - and even then, you'd have to make a case for both building a new telescope and for choosing it over the original outriggers.
Resolution
The angular resolution for a single telescope, to an order of magnitude, is
$$\theta\sim\frac{\lambda}{D}\tag{1}$$
where $\lambda$ is the wavelength of observations and $D$ is the diameter of the dish (adaptive optics, by the way, can make things a bit better). Each Keck telescope has a diameter of $10\text{ m}$ and observes at the optical and near-infrared ranges - so about $4\times10^{-7}$ to $4\times10^{-6}\text{ m}$. This approximation yields angular resolutions of about $0.0083$ to $0.083$ arcseconds. This approximation is accurate to within an order of magnitude.
Likewise, for an interferometer with two telescopes separated by a baseline of distance $b$, the angular resolution is approximately
$$\theta\sim\frac{\lambda}{b}$$
The Keck interferometer had a baseline of $85\text{ m}$ and observed at wavelengths of $2.2\times10^{-6}$ to $10^{-5}\text{ m}$. The approximation gives us resolutions of $0.0053$ to $0.024$ arcseconds - pretty close to the actual values.
What if we add a third telescope, identical to the other two and placed in a line next to them, $85\text{ m}$ from the closer one and $170\text{ m}$ from the farther one? The longest baseline is now $2\times85=170\text{ m}$, twice the baseline of the two-dish setup. We've now made the angular resolution twice as small as before. That's excellent. At $2.2\times10^{-6}\text{ m}$, the resolution is $1.3\times10^{-8}$ radians.
Let's think about observations at $2.2\times10^{-6}\text{ m}$, at the shortest wavelengths the interferometer used. Assume we want to build another telescope with the same angular resolution. Rearranging $(1)$ gives us
$$D\sim\frac{\lambda}{\theta}$$
Plugging in our numbers ($\lambda=2.2\times10^{-6}\text{ m}$ and $\theta=1.3\times10^{-8}$ radians), we find that $D$ should be about $170\text{ m}$ - which is of course the longest baseline of the three-dish interferometer. This is simply implausible at these sort of wavelengths. The largest single-dish optical telescope - observing in shorter wavelengths than this by a factor of about three - is the Gran Telescopio Canarias, at $10.4\text{ m}$ (although the Large Binocular Telescope is slightly better - $11.8\text{ m}$).
If you want to use the interferometer at optical wavelengths, you've still looking at a $\sim55\text{ m}$ dish, and that's currently maybe feasible. There are several extremely large telescopes being planned, notably, the Thirty Meter telescope I mentioned earlier and the European Extremely Large Telescope (E-ELT), coming in at around $40\text{ m}$. The E-ELT should be online as soon as 2-24 (!), which would be fantastic, and would have a comparable resolution to this three-dish interferometer.
Costs
Let's look at how much all of this might cost. The E-ELT is expected to cost a little over one billion euros (1.083 billion, to be precise), which comes out to around \$1-\$1.25 billion, in US dollars (that sounds like a lot, but space-based telescopes can be up to ten times that - though often with higher-quality images). Let's assume that our putative $\sim55\text{ m}$ optical telescope costs \$1.5-\$2 billion to build.
The cost of the instruments for the Keck Observatory ran about \$80 million. This includes the telescopes proper, as well as the cameras and other equipment needed to actually capture the images. Making improvements and other developments has run the total project to about \$200 million (not counting observing costs) We can be generous and say that building a third telescope - identical to the other two - and getting the interferometer system up and running might cost \$50-\$60 million. Maybe that's off, but it certainly can't be wrong by over an order of magnitude. At that point, the entirety of this new Keck Observatory has cost only about one tenth of what a single dish would cost.
Yes, it's likely that I'm wrong somewhere above - that I've made an error in the size of the single-dish telescope, or in estimating the total cost. But the discrepancy is still extraordinary.
Here's the problem, though: Nightly observing costs are high. One night using on Keck telescope costs \$54,000. The interferometer uses both telescopes, and, as an article I cited earlier notes,

it's complicated and expensive to link the giant eyes through a system of optical pathways for just a few dozen nights each year.

Interferometry isn't as simply as looking at an object with two different telescopes. It's complicated and costly. If we assume that operating the interferometer would cost around \$200,000 per night (is that gratuitous?), we find yearly operating costs of $\sim$\$73 million - assuming that it's used every night, which isn't likely. But even accounting for all the nights when the interferometer wouldn't be in use, that's probably still more than the estimated yearly costs for the E-ELT (50 million euros, or \$54.33 million). The single-dish telescope - again, operating a shorter wavelengths than Keck - would probably cost less per night than the new Keck Interferometer.
Perhaps the relatively low construction costs of the third telescope would offset all of this in the short-term (it likely would).
Here's the really obvious thing, however, that I've been ignoring: More dish means more light. An interferometer with an $170\text{ m}$ baseline is not the same as a single dish telescope of the same diameter. The three dishes are much, much smaller than the single one. While the resolutions are the same, the size of the dishes really does matter.
A dish with a diameter of $10\text{ m}$ has an area of $25\pi\text{ m}^2$. A dish with a diameter of $170\text{ m}$ has an area of $7225\pi\text{ m}^2$ - a difference of a factor of 289! If you want to make true comparisons between the new interferometer and a large telescope, you'll need more than three dishes. And that will make costs skyrocket.
Just for fun, imagine that we end up having 50 total dishes, with a baseline. If each costs \$30 to \$40 million to build (and here I'm being generously low), we quickly find that building 48 more dishes runs us to \$1.9 billion, or at least the cost of building our $55\text{ m}$ optical telescope.
Okay, so let's again assume that I made an error somewhere, and this figure is too high. Also, if you ended up building this interferometer (Where would you put it, by the way? Mauna Kea's crowded enough!), you could probably make each dish simpler than Keck I or Keck II. You still have to deal with high nightly operating costs - really high operating costs.
Science!
So, why isn't the interferometer up and running? According to NASA, it's because the interferometer's main target, observing certain circumstellar disks, is done:

NASA’s primary goal in developing and operating the two 10-m telescope KI was the characterization of faint dust disks around nearby main sequence stars.

For an overview, see Millan-Gabet (2011). As the authors note in section 3, the interferometer's primary observing wavelengths (about $10^{-6}$ to $10^{-7}\text{ m}$, the micrometer range) encompass the peak emission wavelengths of circumstellar dust. Additionally, the resolution at these wavelengths can resolve features of circumstellar disks.
As we determined before, it's hard to build a telescope of comparable resolution operating in the near-infrared. Remember, our single-dish telescope operates largely in the optical range, although it would of course have near-infrared capabilities, to an extent. Therefore, an interferometer like the Keck Interferometer is our best choice - and perhaps our only choice, in the foreseeable future. However, it's certainly possible to build extremely large telescopes operating at slightly shorter wavelengths, with roughly the same angular resolution (to an order of magnitude, really). Plus, they'll gather more light, and - as I found shortly before posting this, in an excellent answer by Rob Jeffries - limitations of interferometer instruments and atmospheric issues makes them less versatile than large, single-dish telescopes.
TL;DR
There are a few takeaways here:

Adding a third telescope would give you a resolution unmatched at those wavelengths by any other land-based telescopes.
This would likely be a lot cheaper.
However, you'd gather only a small fraction of the light that a much larger telescope would.
To gather more light, costs would skyrocket - as would technical problems.

So, economically and technically, it's really not feasible to add a third disk to Keck - at least, compared to building a single extremely large telescope.

This is a pretty long answer, and I'm guessing I made an error somewhere. I'll go back and do one more check in the next day or so. If anyone finds one, do let me know!