Endless, unbroken lives of prayer are very possible. Through some searching and with the help of some further reading, this is beginning to be clearer and clearer. We have such a Western Christianity that we have removed a lot of understanding from original Christianity, which happens to have its roots in the East. One large difference is in our understanding of heaven and God's presence.
Western Christianity has this understanding which prays things like, "God, please show up tonight. " "God, be with us tonight." "We prayed, and God just showed up that night." We have this understanding of God being in heaven somewhere above the clouds awaiting our demand to show up.
Eastern Christianity; historical, Biblical Christianity has a different understanding of heaven and God's presence. In Biblical accounts of God's presence, OT and NT, God calls out of heaven, but heaven is near. Heaven is "at hand"...this means heaven is present around them somehow. It was the very presence of God at hand, surrounding them. God inhabits the space surrounding us.
This means we are constantly in the presence of God. We do not actually need to ask for (or demand) that God "show up". This means we have only to grope out after God, or as Luis of Granada (an old century Christian) once said, "any raising of the heart to God."
It is possible at all moments of the day to focus our hearts on God and trust that he is always present. This means we no longer have to be preoccupied with making the sacred time, space, and words in order to really experience the presence of God. This means that we only need to be aware of God's presence with our hearts in order to experience God's presence. This means it is actually possible to experience god all times of the day, continually. This means we can experience God even in the mundane tasks of a day. This means we can join with Brother Lawrence in "Practicing the Presence of God". This means prayer for another person can be merely feeling what they feel (as best you are able) while "raising your heart to God". This means we may have been praying all along as long as we were placing ourselves before God.
We cannot put God into this pious little space and time and assume these are the only moments we experience the presence of God. We cannot dissolve God out of our every day experience. He is always present. We only hope to grope out for him and raise our heart out to him more continually.