The word funnest sounds entirely wrong to my ears, and funner sounds even worse. I would recommend not using either. But both follow the regular rules for forming superlatives in English. Should we accept them?
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Our regular resource for all matters of language usage in Australia, the Macquarie Dictionary, notes that -est is used to form the "superlative degree of adjectives and adverbs". In the case of a one-syllable word ending in a single consonant, it's standard to double that letter when forming the superlative: baddest, thinnest.
While the Macquarie lists both those forms (superlative forms are always spelled out explicitly in the Macquarie, unlike past tenses), it does not included funnest. Yet it's clear we all know what it means (because we're used to that regular rule). Why the exclusion?
The answer is that fun was originally a noun, and the use as an adjective came much later. When we are talking about "having fun" or say something was "so much fun", we're describing an activity, an experience.
The Macquarie notes that fun can be used as an adjective, but adds the note 'colloquial'. Because it's a recent and informal usage, it hasn't yet become acceptable to derive the superlative forms of the adjective. It may well do so one day -- language changes constantly -- but that change hasn't become broadly accepted yet.
For that reason, I'd avoid using that form in any professional or formal context right now. Even in a more casual blog post, I'd try and avoid it by rewriting or reworking the offending sentence, because it is going to prove jarring to many readers. Accuracy matters, and so does readability.
Lifehacker's Mind Your Language column offers bossy advice on improving your writing.