What are the power supply voltages that the 555 and 358 are operating at? Also what is the frequency of the 555 and pulse width? Typically the 555 will put out a pulse from near zero to within a volt or so of its power supply voltage. If that is 12 volts, the output might get to 11 or so. Similarly the 358 can only get with a volt or so of its power supply voltages. Now if the 555 is delivering a signal that is 0 to 11 Volts and you are trying to multiply it by two, non-inverting, the op-amp will have to deliver 0 to 22Volts, clearly beyond its range. So, that is the first thing to look for, what is the voltage you are trying to amplify and what is the available voltage that the amplifier can swing. Obviously you have to stay within that range by lowering the gain, lowering the 555 output or offsetting the amplifier so it stays within range; e.g., -11V to +11V or something similar.
A second thing to look for is the frequency and pulse width of the 555. If these are too high or pulse width too short, the amplifier cannot respond. Furthermore the rise and fall times of a 555 are surely much faster than the slew rate of a 358 will allow. This will put slopes on the rise and fall of the wave form. A 358 is not an appropriate amplifier to amplify fast pulses and this sloping make keep the waveform from reaching its peak depending on how wide the pulse is.
Thirdly, a multimeter is not the proper instrument to measure pulses with. You need a scope to see what is actually happening. Good luck. Bob