Pray continually... (1 Thessalonians 5:17 NIV)
People who are obsessed with God, talk with God. They talk with Him all the time. They don't make a show of it - praying so everyone else can see them in prayer. They simply, consistently, talk with Him.
Their prayers are not selfish, either - asking for comfort, lack of pain, or personal gain. They talk with Him about what is important to Him. Their conversations revolve around aligning their minds, hearts, and actions with His picture, design, and plan. They talk with God the way Jesus taught us to talk with Him.
"This, then, is how you should pray: "'Our Father in heaven, hallowed be your name, your kingdom come, your will be done, on earth as it is in heaven. Give us today our daily bread. And forgive us our debts, as we also have forgiven our debtors. And lead us not into temptation, but deliver us from the evil one.' (Matthew 6:9-13 NIV)
Their relationship with God is respectful, knowing the relationship isn't at all equal. They know that when they differ in opinion on a subject, God's opinion counts. They know that when they come up with different plans to approach some problem, God's plan is right. They know they can approach Him and talk with Him without fear, but he is still "Father in Heaven...Holy...Holy...Holy."
Obsessed prayers talk with Him about His priorities, His task list, and His plan for the day - not theirs. They want His Kingdom to come here on earth...so they ask Him to remove distractions from fulfilling their part of His plan: food, water, shelter, clothing, etc. They want to stay focused on Him and only Him. They know that we are sent to go and make disciples of all people, so they ask God to forgive them where they fall short and they forgive people so they can reach out to the very people who wronged them. They never want the devil to win, so they draw near to God, fight the real enemy, and set the captives free.
And pray in the Spirit on all occasions with all kinds of prayers and requests. With this in mind, be alert and always keep on praying for all the Lord's people. Pray also for me, that whenever I speak, words may be given me so that I will fearlessly make known the mystery of the gospel, for which I am an ambassador in chains. Pray that I may declare it fearlessly, as I should. (Ephesians 6:18-20 NIV)