The short answer is: not all the elements of all the characters are ‘radicals’. For example 凹 (concave, hollow) consists according to the dictionaries of 部首 bushu (or radical if you will, more on that below) 凵 and three more strokes that cannot be further analyzed or categorized.
A more complete answer would depend (as Tsuyoshi Ito already indicated in his note) on what you mean by ‘radical’. In my short answer I assumed ‘radicals’ to mean the 214 bushu from the 康熙字典 (Kangxi dictionary). Those 214 bushu have long been regarded as a standard, but variant lists of bushu exist, both from the past and of the present. Since basically bushu form an index system, whether a bushu is listed or not depends on its usefulness for the function of indexing - not as a tool of structural analysis of the characters.
However, since it is possible to map a lot of the parts of characters onto bushu, most characters could indeed be analyzed as consisting of bushu. A lot of the 象形文字 shōkei moji (pictographs) like 木 tree, 目 eye, 馬 horse, etc.) are bushu themselves and need then not be further analyzed as such. Further, the type of strokes that are used to draw a character have been simplified to the point that almost any segment or stroke can be seen as a ‘variant’ of some bushu - particularly if you include one stroke bushu like 一, 丨, 丶, 丿 and 乙 (the last one can potentially be used as a kind of wild card for any longer curved stroke).
To rephrase your question: are there characters that contain elements or strokes that can not be mapped onto bushu? The answer to that would apparently depend on which list of bushu you use, and how liberal you are in using one stroke bushu as wild cards, but very few at most. However, that is mainly the result of the way characters and strokes have been simplified, resulting in strokes and shapes that overlap today, even though they were different at some time in the past. For that reason identifying elements as bushu not always helps to understand why a character looks a certain way. For starters, it does not help understand the bushu that are complex independent characters themselves. Often these are pictographic or symbolic, even though it is often possible to ‘analyse’ these bushu with yet other bushu, for example 貝 (bushu 154) as consisting of 目 (bushu 109) and two dots (bushu 3 and 4) while in fact it's a pictograph of the shell(s) of a oyster or a clam. Or 自 (bushu 132), which can be seen as consisting of again 目 and a dot, but is in fact a pictograph of a nose.
Likewise other characters of which elements can be mapped onto bushu often have a different history as well. For example 去 is usually indexed under 土 (bushu 32) earth (while the remaining stokes could be seen as a variant of 厶 (bushu 28)), but according to Henshall it originally depicted a lid on a rice container.
Coming back to other possible meanings of ‘radical’, the Wikipedia page "Radical (chinese character)" that Tsuyoshi Ito linked to is accurate enough, but potentially somewhat confusing, because traditionally (in the context of Western linguistics) a radical had nothing to do with an index system (as the bushu are) and was used with a meaning that is rather the opposite of the meaning of terms like ‘signific’ or ‘semantic determinative’ (or ‘phonetic’ for that matter). Boltz (1994:67) explains well how historically a ‘semantic determinative’ like for example 牛 to the left of 勿 in 物 was later added to clarify the meaning ‘creature; thing’ and that thus in fact 勿 should be the ‘root’ or ‘radical’ in the original Western linguistic sense of those terms - exactly not the ‘signific’ or ‘semantic determinative’. In the context of Chinese and Japanese characters the term ‘radical’ is therefore best avoided.
In conclusion I'd like to go into the reason that was given for asking whether kanji are only made of ‘radicals’, which was: can a comprehensive list of those radicals be used to scan for the existance of circular strokes in kanji?
The answer to that question is much more emphatic: no way!
First of all for the reason already (perhaps not explicitly enough) stated, namely that even if you limit kanji to ‘regular scipt’ kanji, kaisho, (楷書) and would take the largest existing list of bushu (and perhaps comparable elements, like e.g. elements from the indexes of some dictionaries like the one by Nelson), a kanji written in kaisho could still contain somewhat surpising shapes (especially if you take all kanji into account, not just regular use ones) to which 丿 or 乙 might point, but not predict the actual shape of.
The second reason presumes that you not necessarily would want to limit your search to kanji written in kaisho style. Certainly in the oldest form of kanji that have been found, strongly curved and even perfecly circular forms were possible (here a nice image of an oracle shell featuring several circles). Currently, artists feel free to create kanji inspired by these very old predecessors.
The later seal script (originally written with brush and ink on bamboo) allowed to some extend curved and more or less circular shapes as well. Interestingly, variants of seal script are still used quite frequently. This style may for example be used for their aesthetic value in stamps, or book headings, sign boards1, etc.
Apart from kaisho and seal script, other styles are in use. For example cursive forms, that are used in hand writing or in calligraphy. Designers of typefaces for printing fonts or sign boards take hints from all of those. See a sample of a handwriting font that includes round shapes. Entire books have been printed with typefaces that are very different from kaisho. That would indicate that the rectangular shapes of kaisho are not a standard that is written in stone.
Notes:
(1) (from second last paragraph): A Google search for "篆書" 看板 will show plenty of results, for example on seikeikai.net, this image showing two shop signs and another sign.
References:
William G. Boltz (1994) The origin and early development of the Chinese writing system. American oriental society: New Haven.
Kenneth G. Henshall (1990) A guide to remembering Japanese characters. Tuttle Company: Vermont & Tokyo.
