An excellent example of a reaction which meets your criteria and produces a significant amount of heat is that of the thermal decomposition of some nitrate esters, the most well studied (because of its use as a propellant in small arms ammunition and artillery) perhaps being nitrocellulose (NC).
Because of the relatively low bond dissociation enthalpy of the nitrate ester group in NC, you tend to see very slow but continuous mass loss, resulting from homolytic cleavage of RO-NO2 bond. This yields *NO2 and other reactive or acidic byproducts which are formed as the glucose ring is destroyed. This process is known as intrinsic decomposition.
If NC does not incorporate stabilisers (such as DPA, EC, MC etc. which are able to mop up *NO2 produced by the intrinsic decomposition process), then a second and more rapid decomposition pathway known as autocatalytic decomposition may result. This pathways leads to the production of yet more *NO2 (effectively a positive feedback loop) as well as to an increase in temperature; conditions which progressively accelerate the rate of reaction leading to a situation known as accelerative self-heating. This certainly meets your criteria for a chemical reaction which accelerates over time!
These processes are certainly not just theoretical and it seems that accelerative self-heating was likely to be the initial cause of a series of explosions which took place in Tianjin China in 2015, leading to the tragic death of at least 173 people.
https://en.wikipedia.org/wiki/2015_Tianjin_explosionsR