Any kind of statistical modeling that doesn't fit neatly into an existing "one (meta)model to rule them all" framework such as generalized linear models.
Stan fittings can be made parallel, some models will scale linearly with the data, but in the main you won't find many big data use cases here.
You also can't use Stan for online learning.
Can’t you loop posteriors as next iterations priors to get a system that learns online?
In the case that your model is extremely simple such that your posterior has a "conjugate prior" (i.e. the posterior and prior are the same family of distribution), this sort of loop back is possible. But where this is possible you have no reason at all to use Stan or MCMC since you can just update your posterior directly.
Sampling is a slow approach when there are other alternatives. For example, if you are after OLS regression, you can do the equivalent with Stan but it may be an order of magnitude slower than plain OLS. Further, the calculation of your likelihood function will scale linearly with the size of the data. But adding new parameters will scale exponentially, so you may find that a model with 2 free parameters which takes 10 minutes to fit takes 2 hours with 3 parameters.
A good thing about Stan however, is that it is parallelisable so you can run it on many cores (and it will scale linearly for a good while) and you can also run it on MPI across many machines. Some regression functions with very large matrices support GPUs (although Stan requires double precision to work). So to some extent you can "throw more money at it" to get a result out and it has been used for very big data problems in astronomy for example which however utilised something like 600k cores if memory serves correctly.
Adding new parameters scales as O(N^5/4) in HMC, whereas it scales as O(N^2) in Metropolis or Gibbs. It's quadrature that scales exponentially in dimension. There's also a constant factor for posterior correlation, which can get nasty. I regularly fit regressions for epidemiology or genomics or education with 10s or even 100s of thousands of parameters on my notebook with one core and no GPU.
MCMC or optimization can be sub-linear or super-linear in the data, depending on the statistical properties of the posterior. Some non-parametric models like Gaussian processes can be cubic in the data size, whereas regressions are often sub-linear (doubling the data doesn't double computation time) because posteriors are better behaved (more normal in the Gaussian sense) when there's more data and hence easier to explore in fewer log density and gradient evaluations.
If by "big data", we're talking about too big to fit in memory, that's right. Stan's fully in-memory. Compute can be distributed and GPU-powered for matrix ops, but all of the data and parameters and the core autodiff expression graph need to fit in memory.
For "medium data", Stan's adaptive Hamiltonian Monte Carlo sampling is much more efficient and scalable to complex models and higher dimensions than Gibbs or Metropolis. I'm fitting a Covid prevalence model using a custom trend-following and mean-reverting second-order autoregression model over 400 distinct regions with weekly data that has 5M data points and 10K parameters and adjusts for sensitivity and specificity of various tests taken. It fits in a single thread using MCMC in 24 hours or so, but we can fit the model with variational inference in a couple minutes. Although variational inference often produces reasonable point estimates in bigger data settings, it doesn't reasonably quantify uncertainty. I'm also working on a genomics model for differential expression of splice variants that involves 120K measurements and just as many parameters to deal with overdispersion of biological replicates in a control and treatment group. We're using variational inference and it fits in a couple minutes for the comparitiver event probabilities we need to estimate.