"Game over" read one poster. "System - go away", said another.
The protests appear to lack leadership and organisation in a country still dominated by veterans of the 1954-1962 independence war against France, including Bouteflika.
But the unrest still poses the biggest challenge to the ailing leader and the ruling elite made up of the ruling party, businessmen, the military and security services.
Tens of thousands of people have rallied in cities around Algeria in the largest protests since the 2011 Arab Spring, calling on Bouteflika, 82, not to submit election papers for the April 18 election. The papers were filed on Sunday.
On Tuesday, hundreds of students protested in cities including Constantine, Annaba and Blida.
Bouteflika, in power for 20 years, has not spoken in public since suffering a stroke in 2013.
Young Algerians who are at the forefront of the protests want a new generation of leaders and have few attachments to the old guard.
After a decade-long Islamist insurgency that Bouteflika crushed early in his rule, Algerians generally tolerated a political system that left little room for dissent as a price to pay for relative peace and stability.
But Algeria's mostly young population, almost 70 percent of whom are aged below 30, are agitating for jobs, better services and an end to rampant corruption in a country that is one of Africa's largest oil producers.
Bouteflika's opponents say he is no longer fit to lead, citing his health and a lack of economic reforms to tackle high unemployment, which exceeds 25 percent among people under 30.