Hi Aaron,
yes, there are a few.
The most obvious candidate would be URW Egyptienne, an expanded version of Frutiger’s Serifa. AFAIK, this slab serif has 25 styles (widths + weights), each with obliques. The family ranges from Extralight Extranarrow to Extrawide Bold. Both extrema can’t compete with sans systems like Univers.

URW Antiqua by Hermann Zapf – not a slab – has ~30 styles (excluding Outline styles), plus another ~30 italics (most – but not all – of them actually obliques). The narrowest weight is Regular Condensed, while the widest is Ultrabold Extrawide. There’s also a Super: darker, but not expanded.

Note that for serif typefaces, traditionally and intrinsically, there are aspects that are considered more important than sheer weight count. Optical sizes, for instance. Or grades. Or contrast. In case you’re looking for records, note that Multiple Master typefaces let you generate an infinite number of styles, if you want to.
F

Beaufort.
Condensed, Normal and Extended, in five weights.
(Not very big serifs, but serifs nonetheless!)

Font Bureau has been in the habit of making extensive width+weight ranges in many of its serif display families. You won't find the kind of extremes as in Univers, and especially not the more Extended widths.

Examples include:

Benton Modern Display -- 4 widths in 4 weights, plus 2 heavier weights in normal width
Escrow -- 3 widths in 5 weights (except only 4 weights in compressed width)
FB Moderno -- 4 widths in 4 weights
Rocky -- 4 widths in 4 weights

All of the above examples include italics for all styles.

Whitman Display has 3 widths in 7 weights, but italics only in the normal width. (See also this style map.)

The astute observer will notice that all of these fall more or less into the Modern genre, which is more amenable to this kind of range treatment.

These are some excellent suggestions. You should also check out Minion for a beautiful, extremely flexible and complete serif face.

...these fall more or less into the Modern genre...

Alternatively, Cheltenham used to have a large family, but a lot didn't make it into digital. (Although FB did Condensed.)
Check out McGrew.
It's slab serif, so I'm surprised it's not more popular.

As Florian mentioned, if the contrast offered by a complete family is really what you are looking for, check out Warnock Pro, Arno, Chaparral, Jenson, Baskerville & Garamond. For what it is worth, the URW versions of Baskerville and Garamond do have condensed, wide, narrow etc.

A few others to consider...

PF Centro has thins.

Vista Slab by Emigre.

Freight Text by Joshua Darden of Garage Fonts. This is a huge collection.

With 12 styles and SC and OSF for all, Starling is pretty robust.

@ Nick: Do you plan on eventually providing SC and OSF for Beaufort or has that been put to rest?

I released a Pro version of Beaufort Normal last year (with sc & alternate figures, and extra language support) , but not the Condensed or Extended. That decision was based on the fact that over the previous decade the Normal had sold much more than the Condensed, Extended, or full family.

A number of families mentioned here are indeed extensive. But I wonder if the key to Aaron's initial question is a system of weights and widths -- not just extensive weights (and goodies) or optical versions.

To the examples I mentioned above, let me add H&FJ's Chronicle Display which comes in three widths and several weights: 6 weights in normal, 5 weights in condensed, and 4 weights in compressed width.

@Nick: Good example. I think Matthew's reworking of Cheltenham for the NYT covered most of this ground, but I haven't seen the fonts themselves, so I can't say just how thorough or extensive the style grid is.

Don’t forget my compatriot Luc[as] de Groot’s TheSerif.
Sixteen styles:

http://www.lucasfonts.com/fonts/theserif/theserif/styles/

(With the 4 Office styles it’s 20, so…)

. . .
Bert Vanderveen BNO

And Fedra. If one considers the B-series to be an alternate to A, there are:

Fedra Serif A: 8 styles
Fedra Serif B: also 8 styles
And then:
Fedra Serif Display with another 8 styles

Maybe I should include the Mono, but that is not a strickt Serif, so…

. . .
Bert Vanderveen BNO

Thank you all for so much good information.

@ Florian: URW Antiqua caught my eye. The whole URW Antiqua and URW Egyptienne volume seemed like a very nice deal for the wide range of fonts in each family.

@ Nick: I really enjoy Beaufort. Too bad you aren't making Pro versions of the other 20 fonts. Are the Pro versions and regular versions of the normal width interchangeable? If someone were to get the regular version, set something, and then later on get the Pro version, could they replace it without displacing the copy?
Also, where would I find McGrew?

@ Kent: Rocky was news to me! I enjoy that face a lot. I've always wanted to see Latins that weren't as narrow as Birch or as extended as Wide Latin. Whitman Display's style map was also very helpful.

@ Aa: I enjoy Minion and Warnock Pro (see avatar) and I love Warnock's optical sizes (caption, subhead, display) but they don't have the variety of widths I'm curious about.

@ Christopher: Starling's heavier weights and italics are nice. Must be nice to design type AND and a be a playboy sailor...

@ Bert: I feel like I find more fonts in the Thesis family every day. Did he make them all at once, or has he just been making them for a long time? Are there different widths of TheSerif?

I suppose all this info leads me to another question: Did Frutiger use the numeral naming system on any serif faces, or have other designers? As far as available widths, URW Antiqua and Beaufort really impressed me.

Any faces with more widths than the ones already mentioned?

I understand serifs don't usually withstand the thin ultra condensed and extra black extended treatment, so lately I've been interested to see some. Thanks again for the responses. Apologies for the flurry of @.

David Berlow used a numbering system for Font Bureau's Giza.

First digit is weight (larger number = heavier) and second digit is width (larger number = wider).

Actually it's quite convoluted to follow. The confusion was compounded by the fact that the numbering system for FB's Bureau Grotesque was the opposite of Giza (width first, then weight) -- although that family has now been superceded by the expanded Bureau Grot, with more typical designations for weight and width.

The Giza styles were developed piecemeal for clients and so not all points in the grid are covered.

If someone were to get the regular version, set something, and then later on get the Pro version, could they replace it without displacing the copy?

Yes.

Also, where would I find McGrew?

American Metal Typefaces of the Twentieth Century by Mac McGrew, published by Oak Knoll.
Yeah, I call it "McGrew" because it's so damn comprehensive and authoritative, just like "Bringhurst"!

Florian

"30 italics (most – but not all – of them actually obliques)."

You need to clarify on this point!

Michael

Michael, you can see the whole family here:
The weights of regular width all have both true italics and obliques, while the narrow, extra narrow, wide and extra wide widths only feature obliques. The condensed has neither italics nor obliques.
F

Computer Modern...

please don’t hit me :-)

Adobe's Kepler. Widths from extended to condensed (extended, normal, semicondensed, cn).Plus – all widths have optical sizes.