With long-term plasticity one refers to the phenomen by which synapses are modified by neural activity and these modifications last for long times, a day perhaps of the order of days. This phenomenon is often described in contrast to short-term plasticity, where the response of a synapse has a non-linear component that depend on the history of the presynaptic activity in a short time in the past of the order of hundreds of ms.
Plasticity is in fact a much more complicated subject and involves processes at different levels, at the synapse, neuron, network and whole brain (e.g., homeostasis) level. Given our current knowledge of memory systems in the brain, how long are we able to trace even small bits of the memory of a stimulus we were exposed in the past?
Obviously one can answer the question at the cognitive level, i.e., we can rephrase the question and ask instead: What's the oldest memory I can recall? It is well known that our behavior as adults is a reflection of even the oldest memories of our lives. However, I would argue that this is an inderect effect, namely those events that happened in our childhood had an influence on how and what we experience say in the adolescence, and then what we experience as adults is a reflection of what we experienced in adolescence. One known phenomenon that seems to "elongate" memories in the future is the one of replay, i.e., the fact that our brain spontaneously recalls memories of the past like in dreaming.
In computational neuroscience, we can trace bits of patterns stored in a neural network with plastic synapses by probing the network, e.g., by presenting a noisy version of a training pattern and testing if the network is able to recall that pattern, something like an associative memory paradigm. A statistical tool used to test the scaling properties of synapse models is the signal-to-noise ratio analysis. Using this analysis, one is able to address the question of a memory lifetime by assessing whether the synapses maintained some information of a pattern in the past (the signal) with statistical arguments (i.e., with respect to the "noise" of all the other memories interfering with the old one).
Is anybody aware of any experiment where people tried to trace single bits of memories at a single synapse level? In models, the weight of single synapses is maintained for ever unless other memories interfere with that. In reality, it would be interesting to see whether single biological synapses can maintain information on long time-scales (days? months? years?), even considering protein turn-over, or if its the network dynamics that plays a crucial role in maintaining those memories or refreshing them.
Note that answers that go beyond the synapse level are also welcome, that's why the main title doesn't specify to much details.