How many RNA-binding proteins can simultaneously bind on a single mRNA?
See this paper. They have studied RBP-protected sites in the entire human transcriptome by RNA-protein crosslinking followed by RNAse digestion and sequencing: PIPseq. 

Figure 1 of the paper shows distribution of protein protected sites in RNAs. They also correlate it with different regions of mRNA and its expression.

They show number of protein protected sites (PPS) per transcript but that is not a proper metric in my opinion. The number should be normalized with transcript length so that you get density of protected sites. From figure 4 (see below) you can roughly estimate that average PPS density is close to 0.6 which means that 60% of any RNA is expected to be protein bound. 

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  Figure 4


Other points to be noted:


Highly translated mRNAs will have multiple ribosomes on their CDS and
are likely to be more protected.
Sequestered RNAs in stress granules will also have high density of PPS
Footprint of different RBPs will be different. So number of proteins that can bind to a mRNA will differ between different RBPs.


Further reading:


http://dx.doi.org/10.1186/gb4153
http://dx.doi.org/10.1016/j.cell.2012.04.031
http://dx.doi.org/10.1016/j.molcel.2012.05.021