There are two relevant terms for word subparts, "stem" and "root". A root is the smallest meaning-bearing part of a word which carries the lexical meaning. A stem contains one or more roots, but typically excludes inflectional affixes. The problem is that there isn't a well-defined definition or theory of "stem' which allows you to say whether 'cow' is just a root, or is it a root and a stem (though 'cows' would not be either, though it would contain a root and a stem if you think that there is a stem). A typical analysis of Spanish (or Italian) is that a verb word like comer contains a stem come- and an inflectional ending, and the stem is made up of the root com- plus a theme vowel e. The main argument for reducing come- to com+e is that verbs end in a very restricted subset of phonemes whereas what comes before that vowel is pretty unrestricted in light of the syllable structure and segment inventory of the language. So either you have to say that there is a really big coincidence that Spanish verb roots all end in /a, i, e/, or you have to say that there's a filter-type rule requiring all Spanish verb roots to end in /a, i, e/ (people tend not to like such rules) – or, you say that all roots have to select one of three meaningless suffixes which is added to the root to form a stem.
In a number of Bantu languages, verbs have a very rich structure where a root combines with many different derivational suffixes and then a lot of inflectional suffixes, and there can be many layers of analysis, so that a root combines with derivational suffixes to give you a Derivational Stem, then some tense and negation markers are added to give an Inflectional Stem, then reduplication gives the Compound Stem, then object markers give you another thing called the Macro-stem, and so on. See Downing 2001 "Ungeneralizable Minimality in Ndebele" for example.
If you're asking for an example of a case where a root combines with an obligatory suffix to give a stem but the stem always has to be combined with something else, then you could look at Romance verbs. However, the stem is often the same as the informal imperative, so you'd have to look at a verb like contar with the stem conta- and the imperative cuenta to get an example where the stem isn't the same as an existing word. On the other hand, if you don't mean stem in the sense "root plus some obligatory affix" (thus, where the stem is the same as the root), then Jknappen's answer gives examples from German.
