Perhaps this isn't what you had in mind, but the Compactness Theorem of first order logic, proved by Goedel, fits your "experimental" metaphor quite well, and variations on its theme have led to some major subtopics of logic.

Namely, the Compactness Theorem asserts that if $T$ is a first order theory and every finite subtheory $T_0\subset T$ is true in some structure, then $T$ also is satisfiable.

For example, this theorem can be used to legitimize nonstandard analysis, since we may let $T$ be the theory of all truths of the standard model $\langle R,Z,+,\cdot,0,1,\lt\rangle$, together with all finite assertions $1+\cdots+1\lt c$, using a new constant symbol $c$. Every finite subtheory of this theory is satisfied simply by interpreting $c$ to be sufficiently large. Thus, by Compactness, the entire theory has a model. Any such model will satisfy all the same truths as the standard model (hence the Transfer principle), yet it will have infinitely large integers and so on.

There are numerous other applications of Compactness in model theory, and its use is pervasive. These applications fit your experimental metaphor, since one probes a theory $T$ by underestanding its finite restrictions.

But this is actual compactness (there is a topological space here, usually sublimated, whose compactness is expressed by the theorem), and you had asked for variations on the theme. So let me mention a few variations on the theme of the Compactness Theorem in logic.

• The concept of Weakly Compact and Strongly Compact cardinals, two large cardinals notions, are based on infinitary analogues of Compactness. Namely, an uncountable cardinal $\kappa$ is (strongly) compact if and only if every theory $T$ in the infinitary $L_{\kappa\kappa}$ logic (allowing meets and joins of size less than $\kappa$ and quantification over less than $\kappa$ many variables at once) which is $\kappa$-satisfiable (every subtheory of size less than $\kappa$ has a model) is satisfiable. This concept generalizes your "probing" idea to understand a very large object by looking at its small subobjects, where the concept of large and small is provided by the cardinal $\kappa$. Among numerous equivalent formulations, it turns out that an uncountable cardinal $\kappa$ is compact if and only if every $\kappa$-complete filter on a set extends to a $\kappa$-complete ultrafilter.
• That concept was generalized by Solovay to the supercompact cardinals, a touchstone in the large cardinal hierarchy.
• The Barwise Compactness Theorem is a generalization of the Compactness Theorem in (a much smaller) infinitary logic, using the concept of admissible sets. One understands a complex theory by looking at comparatively simple subtheories.

Perhaps this isn't what you had in mind, but the Compactness Theorem of first order logic, proved by Goedel, fits your "experimental" metaphor quite well, and variations on its theme have led to some major subtopics of logic.

Namely, the Compactness Theorem asserts that if $T$ is a first order theory and every finite subtheory $T_0\subset T$ is true in some structure, then $T$ also is satisfiable.

For example, this theorem can be used to legitimize nonstandard analysis, since we may let $T$ be the theory of all truths of the standard model $\langle R,Z,+,\cdot,0,1,\lt\rangle$, together with all finite assertions $1+\cdots+1\lt c$, using a new constant symbol $c$. Every finite subtheory of this theory is satisfied simply by interpreting $c$ to be sufficiently large. Thus, by Compactness, the entire theory has a model. Any such model will satisfy all the same truths as the standard model (hence the Transfer principle), yet it will have infinitely large integers and so on.

There are numerous other applications of Compactness in model theory, and its use is pervasive. These applications fit your experimental metaphor, since one probes a theory $T$ by understanding its finite restrictions.

But this is actual compactness (there is a topological space here, usually sublimated, whose compactness is expressed by the theorem), and you had asked for variations on the theme. So let me mention a few variations on the theme of the Compactness Theorem in logic.

• The concept of Weakly Compact and Strongly Compact cardinals, two large cardinals notions, are based on infinitary analogues of Compactness. Namely, an uncountable cardinal $\kappa$ is (strongly) compact if and only if every theory $T$ in the infinitary $L_{\kappa\kappa}$ logic (allowing meets and joins of size less than $\kappa$ and quantification over less than $\kappa$ many variables at once) which is $\kappa$-satisfiable (every subtheory of size less than $\kappa$ has a model) is satisfiable. This concept generalizes your "probing" idea to understand a very large object by looking at its small subobjects, where the concept of large and small is provided by the cardinal $\kappa$. Among numerous equivalent formulations, it turns out that an uncountable cardinal $\kappa$ is compact if and only if every $\kappa$-complete filter on a set extends to a $\kappa$-complete ultrafilter.
• That concept was generalized by Solovay to the supercompact cardinals, a touchstone in the large cardinal hierarchy.
• The Barwise Compactness Theorem is a generalization of the Compactness Theorem in (a much smaller) infinitary logic, using the concept of admissible sets. One understands a complex theory by looking at comparatively simple subtheories.