Of the 676 total possible bigrams "there are only seven bigrams that do not occur among the 2.8 trillion mentions: JQ, QG, QK, QY, QZ, WQ, and WZ."
Norvig also produced data for trigrams through 9-grams "by position within word ... and also by word length." Of the 17576 (26**3) possible trigrams, Norvig found 8,653 (see types count in ngrams3 fusion table), so 8923 (50.768%) trigrams did not occur.
English Letter Frequency Counts: Mayzner Revisited or ETAOIN SRHLDCU
Norvig is a Google research scientist. Analysis was based on the Google Books corpus mid 2012. Norvig normalized words (so word=Word=WORD) and then excluded those that occurred fewer than 100,000 times. Thus these 7 bigrams might legitimately exist (e.g., "howzat" per Ken Grace in the comments) but are very rare (unless you are interested in cricket).
Similarly, some bigrams might not occur depending upon what you mean by "English words." Note that some Roman numerals and abbreviations were included (e.g., no Scrabble word contains "qc" but "QC" for "quality control" was common in my technical books). The list also contains some names and some clearly foreign words (e.g., "forschungsgemeinschaft" in the longest words list must have appeared more than 100,000 times, probably due to funding acknowledgements of the German equivalent of the NSF in scientific papers per Christopher Orr in the comments). I'm unaware of an English "dictionary" word that contains "xx" but the bigram appeared 79 million times in "Exxon" (formed in 1972), "ExxonMobile" (in 1999), "XXX" (whether strike-through, placeholder, hardcore sex rating, movie or album name, Roman numeral 30, genetic abnormality, kisses, extra strong, elided XXXL size, poison/dangerous label, ... [and now a top level domain]), etc.
This list excluded all words that contained numbers or punctuation (presumably also therefore excluding contractions, possessives, and hyphenations).
In addition to Norvig's post, Google Books, and the Brown Corpus (mentioned by acattle), see the Corpus of Contemporary American English (or British or International or others) for easy to use tools and datasets that could be used to extract data patterns that might be of more interest. Google and Microsoft also provide tools and ngram datasets based on web content. Wiktionary might also be useful when considering specific bigrams. Scrabble, word finder, or other game aids might be useful as well.
Note that this answer addresses the limited question of "two [and three] letter combination[s] ... not found in any English words" but not how best to identify "English words" mixed with other text (or vice versa).
Edit: Primarily added Trigram count, links list, and last paragraph.