Are there words that have no plural counterpart, because they are, in fact plural? Words like rice or scissors come to mind.
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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”; “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.
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…
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.
Invariant count nouns
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.
The next categories are singular in form (morphology) but plural-ish in meaning or verb agreement.
(plural meaning, singular form, singular verb agreement)
Discussed in other answers. English has many non-count nouns (such as water), and they are grammatically 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 this is only possible by turning them into count nouns, which changes the meaning to "types of rice, types of gravel, types of furniture," and even with this semantic shift, the word may still sound awkward. (I can't think of any natural context for using furnitures, for example.) So you could say that these words lack plural counterparts.
(plural meaning, singular form, 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. This issue has been discussed in other questions (Is “staff” plural?) Some of these words are commonly used in the plural as count nouns (teams, governments) but others cannot easily be used this way (?polices, ?staffs).
The next categories are plural in form (morphology) but singular-ish in meaning or verb agreement.
(singular meaning, plural form, plural or singular verb agreement depending on the word)
Some words in English are plural in form, but refer to a "single" conceptual object, and consequently are not normally used in the singular. These are often called pluralia tantum (singular: plurale tantum.)
There are several types. The first refer to objects with two noticeable parts, such as pants, trousers, scissors, pliers, tongs, tweezers, spectacles, glasses, eyeglasses, sunglasses. Words like this take plural verb agreement. These nouns are generally not used in the singular by themselves. 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. Others apparently do not, such as glasses (for which glasses case seems to always be used instead of glass case) and clothes (which has changed significantly in meaning and pronunciation from the singular cloth). Here's another question about this class of words: Does (or did) “a trouser” or “a scissor” have a meaning?
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.
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.
Foreign words where only the plural form has been established in English
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.