Standard Mandarin's monophthongs in the usual five vowel analysis /i, u, y, ə, a/ (Pinyin i, u, ü, e, a) [and even if you include the apical vowel /ɨ
/ or /ɹ̩~ɻ̩/ or /z̩~ʐ̩/, Pinyin i] mean that /o/ and /e/ are only present in diphthongs /ou̯, jou̯, wo, je, ɥe, ei̯, wei̯/ (Pinyin: ou, iou, uo, ie, üe [xue, jue, que, yue], ei, uei [written as wei or ui]), bar some interjections, which are generally analysed as lying outside the normal system of phonology.
The existence of /ʊ/ as a monophthong is more debatable; it only occurs with the velar nasal /ŋ/ as a final. In Pinyin, it is written ong and thus analyses it as a pairing with /o/, making /o/ a monophthong; however, the five-vowel analysis follows Zhuyin (ㄨㄥ) and analyses it as /w/ + /ə/ + /ŋ/ (while still contrasting with ㄥ -eng). The notation /ʊ/ seems to be the preferred IPA allocation, giving yet another monophthong vowel.
Another example of a diphthong-only vowel is the case of /ɛ/, which only occurs in /jɛn/ and /ɥɛn/ (Pinyin ian and üan as in xuan, juan, quan, yuan). However, there is considerable variation in the realisation of this phoneme [æ ~ e].
Cantonese has fairly clear tense-lax distribution for certain monopthongs, which could be construed as producing 'new' vowel phonemes, e.g. /o/ only found in /ou/ and derived from mononphthong /ɔː/, /e/ only found in /ei/ from /ɛː/.