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which words don't change when they become plural?

what about cake?

I assume you are thinking of words like "sheep", but there are actually VARIOUS types of words which have no 'singular vs. plural' form.

To start with the more 'standard' sort -- like most nouns, EXCEPT in not adding a plural ending for a plural use:

Quite a number of these are types of animals.

1) many HERD animals (though not cows, pigs, goats)
sheep, swine, moose, deer, reindeer, caribou, elk, antelope, buffalo,bison

head, when used to mean "animals in a herd"; fifty head of cattle

2) '"fish" and many types of fish:
bass, cod, salmon, trout,tuna, halibut, perch, pike
shrimp

3) certain sorts of birds - quail, grouse

others: offsrping, aircraft, cannon

For animals, note that many of these ALSO have regular plurals, though these and the 'singular = plural' are used in differrent ways. Generally, plurals refer to several species or kinds of animal, while the unmarked plural is used to describe multiple individual animals. So, for instance, one would say the order [classification] of fishes, but five fish in an aquarium.

http://en.wikipedia.org/wiki/English_plural#Irregular_Germanic_plurals
http://www2.gsu.edu/~wwwesl/egw/pluralsn.htm

Add to these, names of some people groups:
Those that end in -ese take no plural form, e.g., Chinese, Japanese
Most names for Native Americans: Ojibwa, Iroquois, Blood, Mi'kmaq
Swiss, Québécois. Kiwi
http://en.wikipedia.org/wiki/English_plural

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Then there are nouns which only appear in the plural. The term for this is a "plurale tantum". At first it may seem that these don't perfectly fit your category -- 'the same in singular and plural'-- but they do fit in the sense that ONE set of MANY sets use the same term.

Wikipedia has a list of English pluralia tantum, which I've edited a bit and added a handful:
accommodations, acoustics, alternate angles, amends, annals, archives, arms (weapons), bagpipes, barracks, bellows, billiards, binoculars, bowels, intestines, brains (intellect), chassis, chops, clothes, communications, congratulations, contents, corps, credentials, crossroads, customs, data (in common use), dynamics, earmuffs, faux pas, furnishings, gallows, (sun)glasses, goods, graffiti, (colloard/dandelion) greens, handcuffs, headquarters, jeans, knickers, manacles, mathematics, means, measles, mumps, nuptials, pajamas, (under)pants, panties, pliers, premises, rabbit ears, rompers, scissors, series, shackles, shades, shingles, shorts, species, spectacles, spoils, stairs, strides, thanks, threads, tidings, tights, trousers, tweezers
http://en.wiktionary.org/wiki/Category:English_pluralia_tantum

Many of these, like the animals singular/plurals, can be subdivided into categories. For example:
* Things that come in pairs or have two parts that don't appear alone:
.... Tools: glasses, scissors, binoculars, forceps, tongs, tweezers
.... Clothes: jeans, pants, pajamas, shorts, trousers...

--------------------

There are also (or 'uncountable nouns)'. These are different from "count nouns" like "sheep", in that they do not represent distinct objects (you can say 'one sheep, two sheep' but not one or two furniture(s))

Examples:
* Abstract nouns
.... goodness, idleness, wisdom, deceit, honesty, freshness
* Arts and sciences (even those ending in ics are treated as singular)
.... chemistry, geometry, surgery, biometrics, mechanics, optics, blues (in music)
* Other mass nouns, such as chemical elements and substances:
.... antimony, gold, oxygen, equipment, furniture, species, distress, sand, water, air, information, stuff

At least some of these have been called "singularia tantum"
http://en.wikipedia.org/wiki/Plurale_tantum

Note that SOME words may in certain uses function as 'mass nouns' in others as 'count nouns. "Cake" is a good example ("fruit" is too, though a bit trickier). You can say "I ate too much cake", "I had two PIECES OF cake/fruit" (mass noun) or "he baked two cakes" (as count nouns).

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