There are two limitations: One is the formal number of Dimensions and the other is the memory aspect.
The formal number of dimensions depends on which chart. For example, a bar chart can have 3 Dimensions and 1 Expression (or 2 Dim and several Expr). A pie chart can have 2 Dimensions and 1 expression. But a pivot table can have any number of dimensions and any number of expressions.
The memory aspect is worse. A chart is a cube, which means that the memory usage increases exponentially with the number of dimensions. How many dimensions you can have depends on how many distinct values they have. A simple rule of thumbs is: Never more than 4 dimensions.
Chart Type Dimension Expression
Bar Atmost 2 no limit
Line only 1 No limit
Pie Atmost 2 only 1
Gauge no dimension no limit
Pivot No limit No limit
Straight No limit No limit
Grid Atleast 3 only 1
Radar Only 1 No limit
Mekko Only 1 No limit
Bloack No limit Only 1
Funnal Only 1 Only 1
Scatter No limit Atmost 2 and 3rd expression decide the size of symbol
Combo Atmost 2 No limit
To find the largest number of dimensions, you need to understand the concept of "Dimensionality". See Chart Dimensionality.
For example, a bar chart with two dimensions and one measure has dimensionality=2. But the same is true for a bar chart with one dimension and several measures. So basically, you can calculate the dimensionality using
= NumberOfDimensions + If(NumberOfMeasures>1, 1, 0)
Further, there is a difference between what is possible and what is sensible. The software allows a pie chart with two dimensions and one measure (i.e. dimensionality=2) but in my experience it is not sensible to create such a thing. The user should understand what he is looking at...
With this definition, the table looks like