I have a client who's perception is that the pics I've given her are 'washed out'.  That's fair. She would like them to look "richer".

I would say that in many cases these days, we are purposely giving out images that constrain saturation, and have a longer DR, which can make them seem 'washed-out', especially compared to what we used to give out when we shot film.

But... how can you go richer without over-saturating? For me, even 6-10+ in saturation in PS goes too far. Vibrance doesn't really get there either.

To me, richer means "more color overall, with some extra depth (but not clipped), and not over-saturated".

To me, a richer print doesn't necessarily mean higher saturation. It means
strong blacks and good, maybe a little higher contrast. An 'S' curve will
usually raise the saturation a bit.

I would say choice of hue is as important as saturation and luminance in achieving what most people will think of as "richer".

For example, using gem tones as opposed to pastel or earth tones (hell, even the term "gem tones" denotes "richer").  Obviously if you change saturation and luminance too much, you'll negate the effect, but local changes can do a lot to mitigate the look over all.

So for me it would all start with the art direction and lighting.  Then the post work would contribute to the notion.  I like more contrast than many others, but you don't have to feel the same way.  You can utilize localized contrast control if you don't like an overall contrasty image - thereby creating "richness" where you need it.

If the images are for display, then increasing saturation and contrast is a fairly safe practice, and the only physical constraint is the sRGB gamut. But if the files are destined for print, things can get a bit nasty if you try to push the saturation and contrast a bit too far without checking hard proofs.

Push the blacks  so they are stronger only and this way it won't over saturate your image and will give the perception of a richer contrasty image without affecting colors.

What they said ... but also find out if your client is viewing the pictures on a bad monitor. If you're talking prints, that gives you something to test.

From the time I got started back in the 60s, photographers have been editing for things to look like what they see in magazines, not realizing that magazines don't have anything like the dynamic range of prints. Now, I'm seeing all sorts of "fashionable" edits that resemble the low-contrast reproductions of the 50s and 60s.

Introducing a solid black and a solid white into the image will often bump your contrast. Either curves or simply using the midrange slider in levels will give a richer look to images.

I've tried a global combination-

-adjustment layer set to multiply at 15% (makes the image a little darker)
-curves adjustment layer, slight contrast curve, set to softlight  (contrast, pop)
-another curves layer to negate the shadows from clipping because of the new adjustments
-a copy of the background move to the top, set to color blend mode, 70% (keeps the saturation from getting out of hand)
- and then a vibrance adjustment layer with a slight increase

Overall, it kind of works, but the whole image gets warmer then I'd like, and warmer then I remember. I believe the room was slightly warmer then smoke, so fairly neutral with a faint hint of warmth because of some accent lighting.  It also had black and dark brown furniture, with off-white walls and trim (not off-white as in warm, but off-white as in "not white", but not really grey either). I think that color of paint, combined with incandescent or florescent bulbs in recessed ceiling pots, combined with blue daylight coming in... tempted me to use desaturation on the paint because 5500 window light, and 2600-3000 degree bulbs, makes for some pretty odd color. 

Even though I use strobe to get neutral color, rooms that are strongly lit by other sources as well are tough to get to true color, vs perceptive color because of ambience. The paint is an actual color, but that's not what it looks like once it shifts due to available ambience, and not necessarily what the camera records. Using a white balance card is only accurate to the portion of the room it resides in. About 4300K is maybe the median, but it might only be accurate for 30% of the room.

What I generally do is shoot 3 frames-
-a multi-strobed frame for balanced color and modified contrast  4700k
-an ambient frame for incoming daylight 5200k (natural light)
- an additional ambient frame for interior fixture lighting 2900K-ish

Using all three, I should be able to work on color independently for each condition, relative for the portion of the room it affects, but sometimes this stuff throws me for a loop. I've noticed quite a few prominent architectural photographers are displaying under-saturated portfolio pics as well, so maybe it's the norm when you're working with so much variety of source light these days (especially if the budget doesn't include new bulbs for the whole building/house).  I frequently have 3-5 colors of source light these days. It used to just be daylight and tungsten, then some florescent crept in, and now you got all that plus LED and only god knows what other flavors. Even incoming daylight isn't consistent.

In any case, attempting global richness is difficult in this situation. 

Sometimes I wish somebody would invent a small handheld scanner or a camera app where you could place it on any item and get a numeric color code to work with later.

generally I set curves to luminosity blend mode and deal with
saturation separately

If we were to compare it to models, we'd say "take this albino chick, give her dark brown contacts, put her in a white room, and make it have a certain "richness", oh... and keep it real").

Ask your client to give a comparison, by pointing out something similar to what you've given, and something that way s/he wants it. You will then know for sure, what "richer" means [to your client] (i.e., what the client wants).

Most of the time, when I use the term "rich" to describe or comment on a photograph, I'm referring to the tonality of a b&w image.

I use the word to refer to a photo that displays a wide range of tones with subtle gradations between the tones. Usually "normal' contrast, I suppose, rather than high or low contrast. But the main thing is the subtleties and nuances of the tonality.

"I can see where it could stand a bump in contrast and maybe saturation, but a few different things come to mind when I hear the word "richer". Could you send me a couple of reference images so I know just what you want to see?"

As AKMac said, not necessarily you need to boost colour/saturation, you can gain this by applying selective contrast and a good use of lighting.