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The VIX is a portfolio of OTM options on the SPX with non-zero quotes.

From CBOE white-paper:

Only SPX options quoted with non-zero bid prices are used in the VIX Index calculation. [...] As volatility rises and falls, the strike price range of options with non-zero bids tends to expand and contract. As a result, the number of options used in the VIX Index calculation may vary from month-to-month, day-to-day and possibly, even minute-to-minute.

I know that the number of options changes all the time (with the weights), but I was wondering how big can this number get in practice.

NOTE: I am not interested in the feasibility of such replication or to question any economic benefits in doing it. I am working on a research thesis and I could not find any sources that give an estimate of such quantity.

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    $\begingroup$ Today 2019/11/13 the special VIX calculation for the settlement of VIX futures involved 185 different S&P strikes ranging from 2150 to 3400 which expire on 2019/12/13. 133 were puts and 52 were Calls. This is not the same as the normal VIX calculation, but it gives you a rough idea. markets.cboe.com/us/futures/market_statistics/… $\endgroup$
    – Alex C
    Commented Nov 14, 2019 at 1:29
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    $\begingroup$ "how big could it get?" If every strike between 2150 and 3400 inclusive was used there would be 251 strikes. $\endgroup$
    – Alex C
    Commented Nov 15, 2019 at 18:21

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Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2.

Andersen, Bondarenko & Gonzalez-Perez (2013) identify a new error source in VIX that generates a significant number of jumps in the volatility index unconnected with the underlying volatility process and that weakens the function of VIX as an annualized fair volatility index. This error source is related to the truncation error reported in Jiang & Tian (2005) but differs because it is generated by the CBOE rule that determines the minimum and maximum strikes considered in the volatility index formula (cutting-wings rule). After Andersen, Bondarenko & Gonzalez-Perez (2013) identify this additional error component, the CBOE added in the VIX white paper the following disclaimer: “as volatility rises and falls, the strike price range of options with nonzero bids tends to expand and contract. As a result, the number of options used in the VIX calculation may vary from month-to-month, day-to-day and possibly, even minuteto-minute.” Nevertheless, some adjustments or changes in the VIX formula should also be considered to reduce this deficiency. The literature basically suggests (i) to make the range of strikes economically invariant and compute the VIX as a Corridor Implied Volatility (CIV) index (see Andersen & Bondarenko (2007), Andersen & Bondarenko (2010), Andersen, Bondarenko & Gonzalez-Perez (2013)), or (ii) extrapolate option prices in the tails using implied volatility functions. Nevertheless, mispriced deep OTM options difficult the success of the extrapolation exercise.

EDIT

After finishing my Thesis about the subject I can provide one further interesting point: The Quality of the VIX calculation depends on the spacing of the strike prices, in terms of standard deviation of price (sd). Jiang & Tian (2005) (https://doi.org/10.1093/rfs/hhi027] find that the errors due to spacing are negligible below 0.35 sd. Regarding the range of strike prices, they find the errors due to so-called truncation of the range to be neglible beyond two sd above and below the at-the-money strike price.

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