The 555 timer can be used as a stateless logic element, with the caveat that the threshold voltages of both inputs are different. This means reduced noise margins on input!
The necessary nugget of information comes from a footnote to Fig.8.2. Functional Block Diagram in the NE555 datasheet:

NE555's inputs are prioritized: RESET has priority over TRIGGER, and TRIGGER has priority over THRESHOLD.
Thus, the truth table looks as follows:
RST|TRG|THR|OUT
---|---|---|---
0 | x | x | 0
1 | 0 | x | 1
1 | 1 | 0 | NC
1 | 1 | 1 | 0
Above, NC means "no change": the previously latched state is unchanged.
This is also shown in the datasheet, table 1:

To get rid of the latch behavior, THR has to be set high, eliminating one half of the truth table:
RST|TRG|THR|OUT
---|---|---|---
0 | 0 | 1 | 0
0 | 1 | 1 | 0
1 | 0 | 1 | 1
1 | 1 | 1 | 0
We get a circuit equivalent to AND with one inverted input:

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We can also explore the other possible two-input combinations:
RST|TRG|THR|OUT RST|TRG|THR|OUT RST|TRG|THR|OUT
---|---|---|--- ---|---|---|--- ---|---|---|---
0 | 0 | 0 | 0 0 | 0 | x |0=R 0 | x | x | 0
0 | 1 | 0 | 0 1 | 0 | x |1=R
1 | 0 | 0 | 1
1 | 1 | 0 | NC
The first table above, with THR=0, is not very useful, since we already have the same outputs when THR=1, except that the NC case would need to be handled. If NC were somehow forced to 0, we'd get the same table as with THR=1. If NC was forced 1, we'd get OUT=#RST - not a useful gate. The other two tables are not useful either: the 2nd one just passes RST to the output, and the third one has a constant output.
The other two potentially useful truth tables are:
(NAND?)
RST|TRG|THR|OUT RST|TRG|THR|OUT
---|---|---|--- ---|---|---|---
1 | 0 | 0 | 1 0 | 1 | 0 | 0
1 | 0 | 1 | 1 0 | 1 | 1 | 0
1 | 1 | 0 | NC 1 | 1 | 0 | NC
1 | 1 | 1 | 0 1 | 1 | 1 | 0
In those tables, the NC cases would need to be "forced" to 1. The second table just becomes the one we got with THR=1, so it's not useful.
The first table could yield a NAND, if we could force the NC case to 1. That could be done using a diode and a resistor:

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We also get an output inhibit function without additional components, using the reset input:

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For reference, below is the internal schematic of the bipolar 555. The CMOS schematic is equivalent, so the function of all inputs is similar.

Could we use the control input? Tying it to either voltage rail is problematic, since the comparator output states will depend on the offset voltages of their input stages:
- when CONTROL is LOW,
- a low TRIGGER and THRESHOLD is problematic;
- when CONTROL is HIGH,
- a high TRIGGER or THRESHOLD is problematic.
So, one idea to exploit the CONTROL input would be to ensure that the voltage on it doesn't reach the rails.
This can be achieved by using a series resistor between a totem-pole output and the CONTROL input. We can then fix the TRIGGER level, at 2/3 VCC, and use the CONTROL input as-if it was an inverted TRIGGER input, thus obtaining an AND function:

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