The strictest answer to your question would likely be one of the several Grand Viziers of the Ottoman Empire who began as a Janissary. The Janissaries were Christian children who were abducted, converted to Islam and conscripted as imperial soldiers and civil servants. The office of Grand Vizier was for all intents and purposes equivalent to Prime Minister with both civil and military authority. It was open to be held by anyone at the pleasure of the Sultan, including the Janissaries.

But probably the nearest answer to what you're interested in hearing would be John Shalikashvili (1936 – 2011), an American. He was drafted as a private in the US Army and rose to the rank of General and the office of Chairman of the Joint Chiefs of Staff. Both the rank and office were the highest obtainable by a serviceman or woman.

The guy had quite the childhood. He was born in Warsaw shortly before the German invasion of Poland and survived the conquest of his home city in 1939. His family fled to Germany in 1944 as the Soviets approached, and emigrated to the US in 1952 when he was 16. John became a US citizen in 1958 and was drafted two months later, during the early phase of the Vietnam War (in which he was deployed). John began as a private, but spent little time enlisted and was commissioned a 2nd Lieutenant the next year. He served in every level of unit command during his career before being appointed to Chairman in 1993. That office is an advisory position, however, and does not have command authority over either the military or the state, placing General Shalikashvili junior to the likes of Grand Vizier Sokollu Mehmed Pasha.

Although his case lacks the element of compulsion implied by "drafted," the British army offers a single example of a soldier rising from the lowest rank, that of of private, to the highest, that of Field Marshal. This is the case of Sir William Robertson, who entered the army as a private soldier in 1877 and went on to serve as Chief of the Imperial General Staff from 1916 to 1918. This was the highest rank available to any British officer, and of course unlike John Shalikashvili, mentioned in another response, Robertson served at this level in the midst of a world war. He also spent far longer – 11 years – serving in the ranks.

Roberston's extraordinary career was in part a product of his need to acquire qualifications that would supplement his income. He was commissioned at a time when the typical British officer was expected to have a private income sufficiently large to significantly supplement his army pay. Robertson, who was the son of two domestic servants, had no such resources, and so embarked on an ambitious programme of self-education in an attempt to remedy this situation.

From an earlier answer of mine on army pay:

In 1914, basic pay in the British Army was therefore low by the standards of the time. Officers were expected to be gentlemen and were generally expected to have the private means of gentlemen. Officers, especially those in the most exclusive regiments, required private means in order to pay for their own uniforms (several different and often elaborate ones were required), additional equipment and mess fees, plus "civilian clothes made by a suitably expensive tailor, and two riding horses." All in all, these costs were so substantial that "a newly arrived second lieutenant joining a cavalry regiment was expected to have a private income at least greater than his commanding officer's annual salary." In practice, every junior officer in a regular British regiment required a subsidy of at least £100 from his family in order to serve with honour.

Those who lacked such wealth were forced to settle for less costly careers in colonial regiments or the Indian Army. The career of Field Marshal Sir William Robertson, the Chief of the Imperial General Staff during the Great War, and the only man ever to rise through the ranks of the British army from private to field marshal, is a good example of the practical implications of low pay. Robertson began his career as an officer by accepting a posting to India where, because the Indian government offered cash awards to men who qualified as interpreters – a set rate for each language spoken – he used his spare time to acquire proficiency in six local languages, from Persian to Gurkhali, in order to supplement his pay. Lack of a private income plagued Robertson's entire army career; "as a junior officer he drank water in the mess while his fellow officers quaffed champagne, and he would leave the room to smoke his pipe - only cigars were permitted in the mess - as pipe tobacco was all he could afford."

Well, Egypt was ruled for quite a long time by Mamluks, slave-soldiers who were captured as children and sculpted into Islamic soldiers with little loyalty to anyone other than the lord who financed their training. Eventually they seized power themselves, and for a long time, rulers from the Mamluk Sultanate originated as slaves and rose up through the ranks of the Mamluk system.

Drafted implies modern armies, so I'm not sure if this answer is what you're looking for. But if you're willing to extend it to any soldier who was forced in some way to fight, you could point to the Mamluk Sultanate as proof that they ascended all the way to the top and became rulers.

I'll just note that in a large number of western armies during the cold war (the Scandinavian countries comes to mind), the majority of eligible males would be required to do military service - commonly one year or thereabouts - and that the completion of at least a part of this service would commonly be a requirement for acceptance into NCO/officer training. Thus the vast majority, if not all, of the heads of the armed forces (the highest obtainable rank - whatever the title used in any particular case) of these countries during the post-war period - including the current holders of these positions - would have started their careers as private conscripts/draftees.