"Inflammable" is derived from the verb inflame, which comes from in- and flame. The OED identifies the prefix as in-2, indicating the second definition of the prefix, rather than the third, which is the negation which is what you believed it to be from. I quote the right definition below:
used in combination with verbs or their derivatives, [...] with the senses ‘into, in, within; on, upon; towards, against’, sometimes expressing onward motion or continuance, [...] . (emphasis mine)
To inflame something is to set it on fire–i.e. to use motion to cause something to be in flames.
"Invaluable" does come from in- expressing negation, and thus it means not able to be valued. However, this can be interpreted two different ways—one, it is so worthless that it has no value, or two, it is so valuable that we can not put a value on it—like the concept of there being no finite number that is larger than the rest—you can always add one. The common meaning is #2, but the OED recognizes both definitions.
Neither of these examples are exceptions—the first is misleading because the two prefixes look identical, the second can be understood in two separate ways. The best way to figure these out—have a good dictionary at hand.
The OED’s 1701 citation (the earliest) is from Nehemiah Grew’s Cosmologia sacra: or a discourse of the universe. It reads:
It is the advantagious Organism of the Eye, by which that is procured.
At the time, it meant Organic structure.
The 1834 citation is from Philosophical transactions of the Royal Society of London and reads:
The introduction of new powers into an organism necessarily requires a
modification in its mode of development.
The meaning there is An individual animal, plant, or single-celled life form. Also: the material structure of such an individual; an instance of this.
Best Answer
'Fun' and 'fact': the early years together
As JEL's answer details, the pairing of fun and fact as coequal elements in serial lists goes back many decades. In addition to JEL's 1853 example of "Fun, Fact and Fancy," searches of the Library of Congress and Elephind newspaper databases turn up multiple matches for a regular column titled "Fun, Fact & Fancy" in the [Clearfield, Pennsylvania] Raftsman's Journal (for example, in the issue of June 25, 1857), a regular column titled "Fun, Facts and Facetiæ" in the [Towanda, Pennsylvania] Bradford Reporter (for example, in the issue of October 26, 1865) and—somewhat later—a regular column in the "Children's Tribune" section of the New York Tribune called "Fun, Facts and Fancy" (for example, in the issue of July 20, 1919).
Significantly earlier than the 1853 Southern Standard example of "Fun, Fact and Fancy" is one from the New York Herald (May 5, 1849) that flips the order of fun and facts:
And earlier than that is the headline "Facts, Fun, Fancies" in the Cadiz [Ohio] Sentinel (May 28, 1845).
Other early list matches that owe a debt to consonance include "Fun and Folly, Fact and Fiction" in the Evansville [Indiana] Daily Journal (August 22, 1850), "Of Fun, Fact, and Fiction" in the New York Daily Tribune (March 14, 1856), "fact, fun and frolic" in the [Washington, D.C.] Evening Star (April 21, 1862), and "Fun, Fact and Fashion!" in the Lansing [Michigan] State Republican (April 13, 1864),
An advertisement for a campaign paper in the [Warren, Ohio] Western Reserve Chronicle (May 21, 1856) breaks with the alliterative theme by offering a list that includes "facts, fun, and pepper." Likewise, an advertisement posted weekly in the [Oregon, Missouri] Holt County Sentinel (and later in other newspapers) beginning on July 7, 1865, promises "fun, fact and incident" in a magazine seeking subscribers. Yet another instance, appearing in an anecdote from the New Orleans [Louisiana] Daily Crescent (September 20, 1854), refers to "fun, fact and interest."
But even though these many examples establish a close proximity between fun and fact by the 1860s—especially as members of the the trio "fun, fact and fancy"—all of the instances I found involve fun and fact appearing as coequal list items, not as a words forming single term.
'Fun fact' as a single (and popular) idea
An gram chart of "fun fact" (blue line) and "fun facts" (red line) for the period 1900–2005 shows no sustained enthusiasm for either term until the 1970s, followed by a considerable ascent starting in the mid-1980s:
What happened to trigger that ascent? The likeliest culprit, I think, is an advertising feature called "Fun Facts" or "Fun Facts to Know & Tell," which ran for two or three years during the early 1970s in the color comics section of Sunday newspapers throughout the United States. Although the feature was sponsored (by Wrigley's Chewing Gum), it was drawn similarly to "Ripley's Believe It or Not" and "Cappy Dick," both of which presented bits of fresh trivia and amazing factoids in comic strip format every week.
Here is a brief mention of "Fun Facts" in Rick Marschall, "Cartoons and the Selling of America," in Marschall & Bernard, Drawing Power: A Compendium of Cartoon Advertising (2011):
You can see (at least temporarily) an example of Wrigley's "Fun Facts" (subtitled "fun things to know and tell") from a Sunday [Louisville, Kentucky] Courier-Journal & Times published sometime in 1972 here (it's image 3 of the five-image series). Unfortunately, the image appears in a Craigslist listing and is not likely to last terribly long. I copied the image to my computer, however, and if I ever figure out how to post images in answers at EL&U, I will do so with this one.
Wrigley's "Fun Facts" strip was so popular that Xerox Publications published a collection of the strips in 1973 as Fun-to-Know Facts. Google Books lists a copy of this book, but the copy isn't searchable, even in snippet view.
In any event, I have used the wording "fun fact to know and tell" many, many times when dispensing trivia, and I've always considered the origin of the phrase to be the Wrigley's-sponsored Sunday comics feature/advertisement. A Google Books search finds relevant matches from as early 1984. From the "People" page of InfoWorld magazine (July 23, 1984):
Elephind searches of various newspaper databases yield matches for "fun fact[s] dating to the early 1970s. In a letter to the editor of the Columbia [Missouri] Missourian (April 20, 1974) from Bob Stone:
From an advertisement for Schmidt's Beer in the [Ithaca, New York] Ithacan (October 14, 1976):
And in JoAnne Goldberg, "Behrend Trivia," in the [Erie, Pennsylvania] Behrend Collegian (February 17, 1977):
All three of these matches are from college newspapers, which suggests that students may have been responsible for the initial spread of "fun fact[s]." Also of possible interest is Jim Fowler, "Between the Lines," in the [Houston, Texas] Rice Thresher (January 31, 1977):
This instance unites college newspaper writing (the Rice Thresher was a student publication of Rice University) and the writer's still-fresh memory of Wrigley's "Fun Facts."
Elephind also uncovers a very localized but interesting outburst of "fun facts" from the 1950s. Mrs. Raymond Hoy, of Swenson, Texas, published a column in the Aspermont [Texas] Star called "News From Swenson," which several times began with the opening line "Fun. facts and folks" (note the midline period after the word "Fun," a typographical oddity that appears in all three instances of this phrase in 1950 and 1951 editions of the Aspermont Star that Elephind pulls up).
The expression "fun facts" seems to have made a lasting impression in Aspermont, because "Phoenix Club Entertains With Dinner on Saturday," in the February 14, 1957, Aspermont Star mentions that the expression was used as the name of a party game at the club dinner:
Notwithstanding Aspermont's special claim to precedence in the use of "fun facts," it seems likely that this local usage had no wider effect and is unrelated to the rise of fun fact[s]" more than a decade later. (Elephind does not return any confirmable matches for "fun facts" between this 1957 instance and the 1974 instance from Columbia, Missouri cited above.)
Conclusions
Fun and fact[s] have been hanging around together since at least 1845 in the form of contiguous entries in lists of parallel elements. But fun fact[s] as a descriptive term appears to be a much younger expression. The earliest definite sighting of it that I've been able to confirm is from 1972, in "Fun Facts," a surprisingly popular series of comic-strip-like advertisements for Wrigley's chewing gum.
I remember reading that weekly feature in the Houston Post during the early 1970s, and I think (without definitive proof) that it may have been responsible for popularizing the term in the United States. Certainly its use as a synonym for "interesting bit of trivia" is entirely in keeping with the nuggets that the comic-page "Fun Facts" served up every week.
The earliest match for "fun fact[s]" in Google Books searches is from 1984. Elephind locates confirmable newspaper instances from as early as 1974, largely from college newspapers during that decade.
Elephind also scares up several occurrences of "fun facts" in a small-town newspaper in north-central Texas during the 1950s. These occurrences deserve special mention as being the first definite instances of the term's use, but they seem unrelated to the subsequent popularity of the phrase starting in the 1970s.