TITLE: Which integers can be expressed as a sum of three cubes in infinitely many ways?
QUESTION [42 upvotes]: For fixed $n \in \mathbb{N}$ consider integer solutions to
$$x^3+y^3+z^3=n \qquad (1) $$
If $n$ is a cube or twice a cube, identities exist.
Elkies suggests no other polynomial identities are known.

For which $n$ (1) has infinitely many integer solutions?

Added

Is there $n$, not a cube or twice a cube, which allows infinitely
  many solutions?

Added 2019-09-23:
The number of solutions can be unbounded.
For integers $n_0,A,B$ set $z=Ax+By$ and consider
$x^3+y^3+(Ax+By)^3=n_0$. This is elliptic curve
and it may have infinitely many rational points
coming from the group law. Take $k$ rational points
$(X_i/Z_i,Y_i/Z_i)$. Set $Z=\rm{lcm}\{Z_i\}$.
Then $n_0 Z^3$ has the $k$ integer solutions $(Z X_i/Z_i,Z Y_i/Z_i)$.

REPLY [27 votes]: As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.
A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$. 
Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):
(1960s)  

$87 = 4271^3 – 4126^3 – 1972^3$ 
$96 = 13139^3 -15250^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$ 
$80 = 103532^3 -112969^3 + 69241^3$

(1990s)

$39 = 134476^3 - 159380^3 + 117367^3$ 
$75 = 435203083^3 – 435203231^3 + 4381159^3$
$84 = 41639611^3 – 41531726^3 – 8241191^3$

(2000s)  

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$ 
$52 = 23961292454^3 - 61922712865^3 + 60702901317^3$
$74 = 66229832190556^3 − 284650292555885^3 + 283450105697727^3$ 

(2019)   

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$ 
$42 = 80435758145817515^3 - 80538738812075974^3 +  12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = 72054089679353378^3 -74924259395610397^3 + 35961979615356503^3$ 
$165 = 383344975542639445^3 -385495523231271884^3 + 98422560467622814^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).
Numberphile's videos:   

The Uncracked Problem with 33 
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