Which language has the biggest vocabulary?

I would propose a three-way distinction to illuminate the complexity:

1. Vocabulary - Set of words familiar to any one individual speaker
2. Lexicon - Set of words available to a particular linguistic community
3. Dictionary - Complete set of words attested in a language across all linguistic communities

(Note: this is simply for the purposes of elucidating the question of size - not an objective, hard and fast distinction.)

But I would hypothesize that both vocabulary and lexicon would be roughly comparable across similar types of communities over time. So a rural community in modern American will have roughly the same lexicon as a rural community in Ancient Rome. Equally, a community of scholars in a discipline will be operating with lexica of similar magnitude.

If one were to look for a difference, it would be in dictionaries. And here English has the claim to probably one of the largest dictionaries purely by virtue of spanning a number of linguistic communities in many different environments over a long period of time. But the vocabulary of any one individual speaker will be much more limited.

Some of the things people have mentioned here are a bad place to look for size differences:

1. Identifying any one area where one language has more synonyms than another (e.g. English having multiple words for many verbs) - each language has different areas of diversity and redundancy and areas of compression.
2. Looking at typological differences (e.g. German creating new words through compounds vs. English using phrases) - these phenomena only account for very marginal differences.
3. Borrowing or tendency to avoid borrowing (e.g. French vs. Russian) - all languages in contact borrow - or create new words to reflect needs.
4. Homonyms - some languages (particularly those with more limited phonologies/phonotactics may have more homonyms than others but even that doesn't impact on overall size - unless we limit ourselves to counting unique headwords regardless of sense.

Ultimately, this question has no precise answer nor does it really need one. The important thing for a language is its facility to create as many new words as are necessary to its speakers. And in this all languages are the same.