Are there words that have no plural counterpart, because they are, in fact plural? Words like rice or scissors come to mind.

There are several different categories of nouns that might be considered to have no plural counterpart. The concept of "plurality" has several dimensions in English—meaning/semantics (does a noun refer to something made up of many identifiable sub-units?), form/morphology (does a noun have a plural suffix, such as -s?), agreement (when the noun is the subject of a clause, does the main verb take a singular form or a plural form?)—and these can all vary somewhat independently from one another.

Count nouns with a single invariant form without the -(e)s suffix

These doesn't really meet your requirement at all in my opinion, but I thought I'd mention them for the sake of completeness. Some singular nouns like deer have the same surface form in the plural: deer. But the meaning and grammatical agreement are different for the singular and plural forms, so rather than saying that these nouns lack a plural form, I'd say instead that the plural is identical to the singular.

This category of nouns is relatively small compared to the category of nouns that inflect regularly with the plural suffix -(e)s, but in absolute terms, there are a fair number of invariant nouns, and some of them are pretty common. Many of them refer to animals, such as deer, mentioned above, or sheep. (Some nouns can be used invariantly, but also have a regular plural form, like fish-fishes: these definitely wouldn't qualify as an answer to this question.) This category is somewhat productive, specifically in the context of game animals (see the following discussion at the "Safaritalk" forum, "
What is the correct way to describe wild animals in plural form? "Lion" or "lions"?") and loanwords (a number of loanwords from Japanese can be used invariantly, although many of them do also have anglicized -(e)s plurals).

1. Singular form, plural-ish in meaning or agreement

The next categories are singular in form (morphology) but plural-ish in meaning or verb agreement.

1a. Non-count nouns

(singular form, mass-y or plural meaning, singular verb agreement)

Discussed in other answers. English has many non-count nouns (such as water). Grammatically, nouns like water are treated as singular, not plural, even though you might think some of them should be plural based on their meaning. This includes words like rice, gravel, furniture.

It is possible to create plural forms of these words by adding -(e)s (rices, gravels, furnitures) but doing so requires that they be converted into count nouns, which changes the meaning to "types of rice, types of gravel, types of furniture". Even with this semantic shift, the pluralized form may still sound awkward. (I can't think of any natural context where I would use furnitures, for example.) So you could say that these words lack plural counterparts.

1b. Collective nouns

(singular form, plural meaning, singular or plural verb agreement depending on the variety of English spoken)

Some nouns that are singular in form can take plural verb agreement when used collectively to refer to a group of people or animals. This issue has been discussed in other questions (Is “staff” plural?). Some of these words are commonly pluralized and used as count nouns (teams, governments) but others cannot easily be used this way (?polices, ?staffs).

Peter Shor pointed out in a comment that the collective noun cattle, meaning "bovine animals", is essentially never used with singular reference, or singular verb agreement, in modern standard English. (Use in reference to one animal is known to occur dialectally.) The word vermin is also most often used with plural reference and verb agreement, although use for a single animal seems to be possible for some people. It seems to me that the plural forms ?*cattles and ?*vermins are not really used in standard English.

2. Plural form, singular-ish meaning or agreement

The next categories are plural in form (morphology) but singular-ish in meaning or verb agreement.

2a. Pluralia tantum

(plural form, singular-ish meaning, plural or singular verb agreement depending on the word)

Some words in English are plural in form, but not normally used in the singular. These are often called pluralia tantum (singular: plurale tantum), Latin for "plural only".

There are several types.

2a.i Semantic duals

(plural form, dual-ish meaning, plural verb agreement)

The first refer to things that can be conceptualized as single objects, but that have two noticeable parts, such as pants, trousers, scissors, pliers, tongs, tweezers, spectacles, glasses, eyeglasses, sunglasses. The preceding words take plural verb agreement in standard English. These nouns are generally not used in the singular by themselves (or in the case of glasses, the singular form glass is only used as another word with another meaning). However, some of them do have singular forms that may be used in compound words such as scissor-kick (which coexists with scissors-kick) or trouser press. Glasses case seems to always be used instead of glass case. Here's another question about this class of words: Does (or did) “a trouser” or “a scissor” have a meaning?

2a.ii The word "clothes"

Clothes used to be the plural of cloth, but it has changed so significantly in meaning and pronunciation that it is now considered a distinct word that remains plural, but has no corresponding singular form, even when it is used as an attributive noun as in clothes closet. The plural of the word cloth in modern English is cloths.

2a.iii Disease words and other miscellaneous abstract nouns

(plural form, singular-ish or non-count meaning, singular verb agreement)

The next type have the plural suffix -s, but refer to somewhat abstract or formless concepts, and usually take singular verb agreement in modern English. This includes the diseases measles, mumps, rickets, and shingles, some nouns that end in -ics such as mathematics and physics, and the noun news. I haven't found much use of these in the singular even as the first element of compound words (for example, we use the plural in the compounds measles vaccine, mathematics teacher, aerobics class). Here is a Google Ngram Viewer graph comparing the relative frequencies of scissor, trouser, measle and mump.

2a.vi Latin pluralia tantum loanwords

(plural form, singular meaning, unclear verb agreement)

The next words I'll discuss are more obscure. From Latin, the words kalends/calends, nones, and ides are all plural in form, but singular in meaning (referring to specific days in the months of the Roman calendar). Because of this, their names in Latin, from which these English words are derived, had no singular counterparts. Despite this, singular forms such as calend or ide have been attested in English, but they are now considered obsolete. Whether the forms ending with s are treated as plural or singular for the purpose of verb agreement seems unclear; Wikipedia uses singular agreement, but other sources (here for example) seem to avoid this.

3. Foreign words where only the plural form has been established in English

(Latinate plural forms, plural-ish meaning, plural verb agreement)

The word mores (pronounced "MOAR-eaze" /ˈmɔriːz/, or by some people, "MOAR-ayze" /ˈmɔreɪz/) is derived from a Latin plural and seems to always take plural agreement in English. There was a corresponding singular form in Latin (mos), but it is never used in English. A back-formed English singular more is occasionally seen, but is not widely accepted by prescriptive guides to usage.

Another word like this is manes (two syllables), which comes from a Latin plural-only word for the spirits of the dead. The American Heritage dictionary lists a second definition "(used with a sing. verb) The revered spirit of one who has died" but it seems to be less common.

People who maintain the blog of Oxford Dictionaries’ blog have published a list of 12 words — some of which already appear in various comments and responses posted in response to the original question — that not only lists the words but also discusses their origin and how they are presently used. The link should take the interested parties to the original post:

12 Nouns that are always plural.

I hope that the person who posted the question and others would find the information useful.

Note: If you loathe pictures of cats, then you will probably not enjoy the post; however, it definitely provides at least a few additional nouns that are always used in plural.

Your title asks for nouns that are always plural, but your question seems to ask for nouns that have no plural. I'll answer the latter first.

Non-count nouns are very rarely used in the plural. Some examples include butter, electricity, ballet, and indifference. You could say We tested six butters to see which was best for baking, but this is rare, and the others couldn’t be used this way.

(Many proper names are rarely pluralized, just because there’s only one. But it can be done: “The Sun is so large a million Earths could fit inside”; “there are two Americas, not one”; “it would take nine Chuck Norrises to bring down one Bruce Lee”; “conformable as other household Kates”.)

As for nouns that are always plural, there are a few. Scissors and thanks come to mind. You never give somebody a single thank. Some more are listed here. (They missed dregs though.)

Rice is a mass noun, a noun that signifies unbounded amounts, such as liquid, small objects, and abstract or immeasurable concepts.

Scissors, like rabies, is a plural noun that is treated as singular. (Also called a plurale tantum.) The same as some other tools: pliers, tongs, or tweezers; the diseases mumps, rickets, or shingles; or the games darts, billiards or dominos.

OOPS

Due to a comment, I did some more research and since there is no singular form to rabies as it comes from the latin word rabiēs, from which we get the word rabid, it looks like it is also a mass noun not a plurale tantum. That final "s" can get confusing…