What are the smallest possible RSA private and public keys?
Per the definition of RSA in PKCS#1v2.2

In a valid RSA private key, the RSA modulus $n$ is the product of $u$ distinct odd primes $r_i$, $i=1$, $2$, …, $u$, where $u\ge2$.

That makes $n=3\cdot5=15$ the smallest public modulus.

and the RSA public exponent $e$ is an integer between $3$ and $n–1$ satisfying $\operatorname{GCD}(e,\lambda(n))=1$, where $\lambda(n)=\operatorname{LCM}(r_1–1,\ldots,r_u–1)$

That makes $e=3$ the smallest public exponent. $(n,e)=(15,3)$ happens to be a valid public key, since $\lambda(15)=4$ and $\operatorname{GCD}(3,4)=1$.

The RSA private exponent $d$ is a positive integer less than $n$ satisfying $e\cdot d\equiv1\pmod{\lambda(n)}\,$.

That makes $d=1$ the smallest private exponent. It corresponds e.g. to $(n,e)=(15,5)$. Encryption (and decryption) with that key is identity, but there is no worded prescription against that.

If we prohibit $d=1$, then $d=3$ becomes the smallest private exponent, matching $(n,e)=(15,3)$. More generally, different definitions of RSA yield different lower limits. Allowing $u=1$, $e=1$, and removing the prescription that $r_i$ is odd, makes $(n,e,d)=(2,1,1)$ acceptable. For FIPS 186-4, the smallest $n$ is 1024-bit, likelyA $(\lfloor2^{1023/2}\rfloor+257)\cdot(\lfloor2^{1023/2}\rfloor+2^{412}+431)\,$; the smallest $e$ is $65537\,$; and the smallest $d$ isB $2^{512}+1$.

A: Under the plausible assumption that each of $\lfloor2^{1023/2}\rfloor+256$, $\lfloor2^{1023/2}\rfloor+258$,  $\lfloor2^{1023/2}\rfloor+2^{412}+430$ and $\lfloor2^{1023/2}\rfloor+2^{412}+432$ has a prime factor at least $2^{100}$, which I did not check.
B: Some valid matching public key $(n,e)$ exists with high likelihood. It's a mildly interesting problem to exhibit one.