A ridiculous combinatorial cardinal characteristic of the continuum? This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the abstraction right.
A family of subsets of $\mathbb{N}$ is centered if every finite subfamily 
has an infinite intersection. A pseudointersection of a family is an infinite set that is almost contained in every member of the family.
Let $\mathfrak{ridiculous}$ be the the minimal cardinality of a centered family
of subsets of $\mathbb{N}$ with no 2 to 1 image that has a pseudointersection.
By 2 to 1 image of a family $\mathcal{A}$
we mean the family $\{f[A] : A\in\mathcal{A}\}$,
for some 2 to 1 function $f\colon \mathbb{N}\to \mathbb{N}$.
($f[A]:=\{f(n):n\in A\}$).
We know that $\mathfrak{p}\le\mathfrak{ridiculous}\le \operatorname{add}(\mathcal{M})$. (We see this using selection principles; direct arguments of course must exist, too.)
Question. Is $\mathfrak{ridiculous}=\mathfrak{p}$?
A negative answer (i.e., consistently ``no'') would be ridiculous.
 A: The cardinal $\mathfrak{ridiculous}$ is equal to $\mathfrak p$ (which is equal to the smallest character of a free filter without infinite pseudointesection on $\omega$). It suffices to prove that a free filter $\mathcal F$ on $\omega$ has infinite pseudointersection if $\mathcal F$ has a base $\mathcal B$ of cardinality $|\mathcal B|<\mathfrak{ridiculous}$. 
The latter inequality implies that there exists a sequence $(\{x_n,y_n\})_{n\in\omega}$ of pairwise disjoint doubletons such that each 
 basic set $B\in\mathcal B$ (and consequently, each set in the filter  $\mathcal F$) intersects all but finitely many doubletons $\{x_n,y_n\}$. 
If $\{x_n\}_{n\in\omega}$ is not a pseudointersection of $\mathcal F$, then there exists a set $E\in\mathcal F$ such that the set $\Omega=\{n\in\omega:x_n\notin E\}$ is infinite. We claim that the set $\{y_n\}_{n\in\Omega}$ is a pseudointersection of $\mathcal F$. Indeed, for any $F\in\mathcal F$ the set $F\cap E$ intersects all but finitely many doubletons $\{x_n,y_n\}$, $n\in\Omega$, and contains no points $x_n\in\Omega$. Then $F\cap E$ contains all but finitely many point $y_n$, $n\in\Omega$, and so does the larger set $E$, which means that $\{y_n\}_{n\in\Omega}$ is an infinite pseudointersection of the filter $\mathcal F$.
So, in both cases, $\mathcal F$ has infinite pseudointersection: $\{x_n\}_{n\in\omega}$ or $\{y_n\}_{n\in\Omega}$ for some infinite set $\Omega\subset\omega$.  

Remark. This method easily generalizes to prove that $\mathfrak p$ is equal to the smallest cardinality $\mathcal F$ of a centered family $\mathcal F$ of subsets of $\omega$ such that for any $n$-to-1 map $\varphi:\omega\to\omega$ the image $\varphi(\mathcal F)$ has no infinite pseudointersection.
On the other hand, the smallest cardinality $\mathcal F$ of a centered family  $\mathcal F$ of subsets of $\omega$ such that for any finite-to-one map $\varphi:\omega\to\omega$ the image $\varphi(\mathcal F)$ has no infinite pseudointersection is equal to $\mathfrak b$, see Theorem 9.10 in the survey "Combinatorial Cardinal
Characteristics of the Continuum" by Andreas Blass.
