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Clarifying questions:

What is the question I want to answer with this statistic?
What does the statistic mean?
What are the units?
What is the underlying data for this statistic?
What do typical data points look like? What do extreme values look like?
What is the time range?
What does the graph show? How are the x-axis and y-axis labeled? Is the title/legend appropriate?

Questions that challenge assumptions:

Is a higher or lower value better?
Is it in line with my expectations?
What assumptions am I making?
What assumptions did the analyst make? Were the assumptions tested?
Is there any sampling bias? Desirability bias? Selection bias? Survivorship bias? Confirmation bias? Predictive bias? Anecdotal bias?

Questions that examine the evidence:

How strong is the evidence?
Statistical significance vs business relevance?
What is missing? Any missing data? Missing measures of variation? Missing measures of uncertainty?
How strong is the correlation? How can causality be explored?
What would make me feel more/less confident about this analysis?